Is there a word for this family-ish relation?

Not really a flame- but do you have the same issue when people say “my cousin”, or “my neighbor” or “my coworker” ? “My” doesn’t only imply ownership or possession - it also denotes a relationship.

I’m not sure I follow. Marrying someone makes them, in the eyes of the government, society and the almighty god (or whatever), YOUR spouse. And it goes both ways, as you noted, so it’s not like it’s a patriarchy/misogyny thing, it’s just how you refer to each other.
I’m not sure how you’d refer to your husband/wife/girlfriend/boyfriend/cousin/aunt/uncle etc without using some type of possessive pronoun. IOW, how do you introduce the person you married without using the word ‘my’. “This is Jaime, the person that married me” is not only clunky, but doing everything it can to get people to ask you to clarify.

Sure, if there’s some misogynistic subtext, or even the a hint of the person feeling like they ‘own’ their SO, that’s different, but suggesting you shouldn’t refer to your SO as your SO, gets into SJW territory.
ETA, it would follow that you never refer to others like that as well. That is, you never say to someone ‘her? That’s John’s girlfriend’ or ‘that’s my brother’s husband’, right?

I hate referring to myself because it sounds like I own me and nobody owns me.

Thus making Hortence your daughter’s sister. That you call her that already makes it clear that it’s a bit more complicated than “they’re both my daughters”, but I wouldn’t go beyond “they’re sisters through the other side” unless someone really, really has the right to know (medical or legal reasons). The nosy ones will probably assume that they’re sisters on their mother’s side either way, but the complete details are nobody’s business.

The adopted one is Gertrude, and Hortence is not in a direct child-like relationship to the OP since it’s not the OP but Gertrude who has guardianship of Hortence.

I really started to get irritated when my I spent months listening to my friend complain about his wife (they eventually got divorced), talking about how she should act because she was “MY wife” in a highly possessive tone.

I’ve never introduced my girlfriend as “My girlfriend xxxx”. I usually introduced her by name only as I feel it made her stand as an individual. On the rare occasion, I would introduce here as “My better half, xxxx”. If people asked if she was my wife, I answer no, we’re not married, but might hold her hand a bit tighter or move closer to her to indicate she’s more than just a friend. Thankfully we’ve gone beyond the days of “This is Mrs. lingyi” without any sense of individual identity besides that.

I’ve never been married and the introduction may have changed if I were, but especially today where same sex marriages are legal in many states, I find it especially pretentious to say “My wife or my husband xxxx” or make presumptions or inquiries as to the relationship. Thankfully we’ve gone beyond the days of “This is Mrs. lingyi” without any sense of individual identity besides that.

The introduction of relatives changes depending on on the relationship. I would use a diminutive “My” to introduce a sibling or cousin, but a more formal stronger “My” when introducing an elder such as my parents, grandparents, aunt or uncle to enforce their seniority and my respect for them.

Part of this is the underlying formal and informal speech patterns of Japanese (which I don’t understand or speak) which are subconsciously ingrained in me. Ironically a number of my aunts and uncles had nicknames that could only be used within the family (half of them, I don’t even know their given names) and I’ve only referred to them as Auntie or Uncle outside the family.

She’s your daughster.

In the context of Thanksgiving, I don’t think the question arises. Presumably everyone in the extended family knows the relationships here; you don’t have to explain them to anybody, and I can’t see any compelling need around the Thankgiving table to be classifying and categorising everyone into a neat, recognised and easily-named relationship to yourself. Besides, Hortence is mainly there not because of her relationship to you but because of her relationship to Gertrude, and you can safely leave the two of them to work out the language they are comfortable with to name that relationship.

Outside the thanksgiving context, I agree with what others have said. The primary consideration here is what Hortence wants. By your account, she calls Gertrude and her husband “Mom” and “Dad” and it seems they are, in fact, the parental figures in her life, and at least one of them is her legal guardian. Consistently with that, proceed on the basis that she thinks of you as a grandparent and therefore think of her as your granddaughter, unless she indicates that she would prefer something else. If she has a relationship with her biological grandparents (you don’t mention them in your OP) and is uncomfortable with you encroaching on that, then go with step-granddaughter. It’s not technically accurate, but it’s close enough to the substance of the relationship.

This has the added advantage that it avoids adding to your son’s problem by letting him think or feel that you regard Hortence as a daughter or similar, and therefore as his sibling. He evidently hasn’t yet accepted that Gertrude can think of Hortence as her little sister - that’s a whole other problem for you - but he should find that much easier to accept than the idea that you think of Hortence as a daughter.

Obviously different levels of explanation are appropriate in different situations, but my thought was you could say “She’s our daughter’s sister” with a slight emphasis on “our”. If someone said that to me, I’d understand they were half sisters through a different parent. YMMV.

I’m a big fan of glossing over stuff that’s irrelevant. I’ll misrepresent the heck out of relationships to, say, cashiers who are just making small talk. This came up frequently when my daughter was small and I was in a new relationship with a man who wasn’t her father - they’d look at the two brunettes with the red headed child, and ask where the hair came from. I’d just say “My dad”, because no one cares, honestly, and that was enough of the truth for people I don’t know.

Just call her your niece. She’s younger than you, related to you (sort of) but not directed descended from you? Niece is good enough.

Seconded. Niece is a good non-specific catch-all relationship. There are plenty of cultures where aunt/uncle is a loose title for a respected older person. Niece/nephew just takes it in the opposite direction.

All this depends on who you are talking to. If it’s just a curious stranger you can say whatever you want. In this case daughter, niece, or even step-daughter are perfectly acceptable, the specifics of the relationship aren’t important. You can be more specific for people who are developing a closer relationship with the family.

There are also a lot of informal cousins out there. It tells someone there’s a family relationship without any details about how close. There shouldn’t be an assumption that a cousin is a first cousin blood relative, just some member of the family with a less direct relationship than son or daughter.

(bolding mine)
I chimed in to say much the same.

Some ramblings:

In American-style English, there isn’t a succinct term for this kind of relationship. Maybe settle on something like “niece” or “grandniece”; niece strikes me as right because “daughter’s sister” is awfully close to “sister’s daughter”. A niece isn’t necessarily biologically related anyway - if the person is related to your spouse (e.g. your wife’s sister’s daughter is still your niece). So that kinda fits the bill.

Step-granddaughter also works - admittedly the steps appear to have been designed by Escher, but the effective relationship is grandparentish (she calls your daughter Mom).

Another bit of rambling: If you marry someone who has a child from a former relationship, you would refer to that child as your stepdaughter / stepson whether you were involved in raising that person or not.

If your ex has a child later on, you’re not really considered that child’s step-parent, right?

Ignoring the OP: that’s the relationship your wife has with Hortence. Not-quite-a-stepmother, though they could argue that is a possible title.

What does your wife call Hortence? What does Hortence call your wife? Grandmother? Auntie? something else? If that’s been sorted out, then you can take your lead there.

Hopefully Wolfgang can come to terms with the naming and realize it all has nothing to do with his relationship (however fraught it was / is) with Gertrude.

And good on Gertrude for taking in her sister and providing a stable place to live.

Last bit of rambling then I’ve got to go back to work:
If she wants to start calling you Bawpaw (because her sister/mom’s kids do), it’d be nice if you let her do that - it would probably really make her feel welcomed.

I’ve always thought that we should call our relative’s in-laws our out-laws. Somehow, I’ve never suggested it to anyone.

This thanksgiving I was talking to my MIL about my nephew’s wife and I called her my niece-in-law. Everyone was confused.

Human relationships are complicated. My mom went and got herself a couple of grandchildren without asking my sister or I. There’s no blood relationship at all, nor any sort of direct or indirect legal connection, but her relationship to them is definitely the same sort of relationship as between a grandmother and grandchildren. They refer to her as their “godmother”, but she doesn’t, because being Catholic, that term has a more specific meaning to her that does not apply in this case. I refer to them as her “honorary grandchildren”, which mostly gets the idea across.

And sometimes people invent new words for relationships. For instance, in my family, one’s cousins’ cousins on the other side (to whom one is not directly related) are “turkey cousins”, because we joke about the family of such whom we most often visit with being a bunch of turkeys.

My sentiments exactly, as someone with similar relationships in my own family.

Apologies for the sidetrack, but this thread has reminded me: after Patti Boyd divorced George Harrison (amicably), she married George’s closest friend Eric Clapton (with George’s blessing, not that anyone needed it). George then referred to Eric as “my Husband -In-Law.”

Neither my brother or I are married to our respective life partners. My mother calls them (the life partners) her daughter and son out-laws.

Hortence wouldn’t happen to be red-headed, would she?