So, I’ve got a bit of a twisty one, re: What to call a certain person (cousin, second cousin, etc.)
Let’s say my dad gave me up for adoption. He was never married (my mom died early). He gets married, and his wife’s cousin has a kid.
Now, you’d think the question would be “What does that make the cousin to me, relation-wise?” But it’s one step more complicated than that. The question is: Since I got adopted, what does that make that kid in relationship to my adopted mom?
In speaking, the closest you could probably get that adequately conveys the actual biological and social relationships would be “my biological father’s wife’s cousin’s kid” or if you consider your dad’s wife to be closer to you than that “my biological parents’ cousin’s kid” but that second one 1) makes it seem like both the wife and therefore the cousin and the kid are biologically related to you (they aren’t), and 2) that your father’s wife is your mother in some sense - neither of which you may want to imply.
For your adoptive mom, it’s “my son’s biological father’s wife’s cousin’s kid.”
If you have a relationship with this person and would need to refer to them casually in passing, most Americans end up with “cousin” as a utilitarian “someone I know of as a relative, but related and known through parents/grandparents generation” regardless of the taxonomic particulars, and “cousin once removed” brings the implication that either the person being discussed is younger than you, or that there are very large families involved, depending on the particulars.
For your adoptive mom then, it would be “my son’s cousin/cousin once removed” and people don’t need to get all hung up on the biological particulars.
From what I can follow, there is no real relationship other than the kid and your stepmother. Neither your dad, nor you, nor your adoptive mom are related by blood to the child at all.
You could say your stepmother’s first cousin once removed, if the parent of the kid was a first cousin to your stepmother. I guess your adoptive mom could say the kid was her son’s stepmother’s first cousin once removed, but just saying a connection by marriage would cover it.
Your dad’s (current) wife - “my cousin’s child, hence first cousin once removed.”
Your dad - “my wife’s cousin’s child, hence…uh…?”
(not sure what the terminology is - “first cousin once removed-in-law”?)
For you - “my father’s wife’s first cousin once removed.”
For your adoptive mother - “some kid related to my adopted son through his biological father’s marriage.”
Even the European nobility who tracked all sorts of relationships don’t really have names for this. Maybe that’s because all the relationships they dealt with had much closer blood ties…
I do note it’s not unusual to drop the in-law distinction; your wife’s siblings children are spoken of as your nieces and nephews, not as in-laws. If this carries over to distant cousins, it can get misleading.
What you are asking about is really just genealogy terms which most people don’t care about in casual conversation except maybe for the close relationships.
Colloquial Southern U.S. English often simplifies the nomenclature as completely as possible. People that are about your age or younger that you feel you are related to are ‘cousins’. People that you feel you are related to but significantly older than you are ‘aunts’ or ‘uncles’. It doesn’t really matter what the biological relationship is if there is any at all. Even very close family friends can be ‘cousins’, ‘aunts’ or uncles’ (particularly the latter two).
It keeps things simple and you don’t have to break out a family tree and a projector if you need to make such a reference. However, most of the previous answers are correct if you want to use the proper genealogy terms.
You can diagram lineage. All cousins are in a horizontal line, the same generation as you are. All cousins have a common ancestor and if the degree of any 2 cousins is the same, the ancestor is the same number of generations from each of you.
You and your first cousins have a common grandfather. You and your 2nd cousins have a common great grandfather. You and your third cousins have a common great-great grandfather. To find the degree of cousinery (?), add 1 to the number of greats in the common ancestor’s name.
e.g. If your and your newly found cousin have a common great-great-great grandfather, you are 3+1=4th cousins.
Just a common grandfather (no greats)? 0+1=1st cousin.
If the common ancestor is your g-g-g-grandfather, but only their g-g-grandfather, they are removed from that generation line. They are your 4th cousin once removed.
Even in northern U.S. there were several people I referred to as “Aunt” and “Uncle” back in the 1970s who were no blood relation at all - they were very close friends of my parents (mere acquaintances of my parents were called “Mr. <last name>” or “Mrs. <last name>”); also as you suggest, I had relatives who were genealogically my cousins who were closer to my parents’ age than my age - so they were “Aunt” and “Uncle,” too.