They certainly may and do if English isn’t their first language. There are plenty of languages that do not differentiate terminologically between siblings and cousins. I could well imagine someone who speaks one of these languages carrying the ambiguity over into English.
I am quite close to a second cousin once removed. I’ve always called him “Cousin Fred,” in the same way I call other relatives of his generation “Uncle Sam” or “Aunt Rubella.” I call him that because my grandmother (his first cousin once removed) called him that. My own generation I just call by their first names, no “cousin.”
“Second cousin” in colloquial speech means “cousin once removed” as well as “second cousin.” Because this is ambiguous in a very unhelpful way, people trying to be precise about relationships don’t use “second cousin” this way. It’s pretty common in ordinary speech, though, common enough that I don’t think you can say that it’s wrong. The Oxford English Dictionary lists “(also more loosely) a child of one’s first cousin, one’s first cousin once removed” as a subsidiary meaning.
This is a little bit of a hijack - but when people ( in general, not specifically Dr.Drake) refer to using cousin as a title in the same way they would use “aunt” or “uncle” for others in the cousin’s generation , are they really talking about generation or does it really have a lot to do with age?
My mother is from a large Italian family - she has forty first cousins. And because there are so many of them, the generations almost overlap - the youngest of the cousins is only a year or so older than the oldest of the cousin’s children.* My mother’s cousins range in age from 96 ( if he were still alive) to 67 and they are in the generation that I would call aunt/uncle. The oldest of my generation is 66 - would those who address someone as “cousin so-and-so” because they are of the parents’ generation do so even if the cousin was approximately the same age as the speaker?
* Not even gonna get into uncles/aunts younger than their nieces/nephews.
I think it’s more age, you’re right. But also circumstance.
As an adult, I got to know my grandfather’s cousin, i.e. my first cousin twice removed. Said cousin is about ten years younger than my father, but only about 5 years younger than Cousin Fred, and we met rather coincidentally through a mutual friend (several thousand miles from where she and my grandfather grew up: small world). Because the friend introduced us, and because she’s about 40 years younger than my grandfather, I got to know her by first name only, no honorific. Her mother, whom I knew as a teenager, was always Aunt Olive, though, because that’s how my grandfather referred to her.
Families are complicated!
I have to ask this. In most places in the United States you have to be no closer than second cousins to get married. First cousins once removed would still be too close.
What if a judge mistakenly thought first cousins once removed were second cousins, and issued a marriage certificate? Would you still be validly married? What no one ever found out? What then? 
Cousin marriage law in the United States - Wikipedia suggests that would only be a problem in 8 states (it’s fine in the other 42). This also says that first-cousin marriage is valid in 30 states, including California and New York, so I question your “in most places”—maybe in about half the places, by population.
That wouldn’t happen - when degrees of relatedness are important , governments have handy charts/family trees to tell you who is who. I used to have to determine whether kinship foster parent applicants were “relatives within the third degree” . No way was I trying to determine that without a chart. But if a license was issued that shouldn’t have been, the marriage would be void- but nothing would happen if no one ever found out.
From the OED:
cousin, n. and adj.
A. n.
1.
a. Any collateral relative more distant than a brother or sister; a (distant) relation. Now frequently with modifier, as distant, remote. […]
b. spec. A child of the brother or sister of either of one’s parents; a person with whom one has one or more (typically two) grandparents in common […]
c. […] second cousin is sometimes used loosely to denote a child of one’s first cousin.
So technically your father is right. Colloquially, everybody else is, too.
I worked out a couple of formulas several years ago to determine the degree and removal of a cousin. Standard disclaimers apply.
Number the generations sequentially, ascending from older generations to younger.
gen(x) is the generation number of person x.
Persons p1 and p2 are (Degree) cousins (Remove) removed where
- Degree = min(gen(closest common ancestor)-gen(p1),gen(closest common ancestor)-gen(p2)) - 1
- Remove = abs(gen(p1)-gen(p2))
Using this, you and your siblings are zeroth cousins, and you are your own (-1) cousin. You and your parents are zeroth cousins once removed.
Another example of a family barbecue?
1722? Sounds more like jerky than barbecue…
Relatives who are jerky … you’ve met some of mine I guess!
Same - I’ve had email contact with someone who’s my 13th cousin once removed. The once removed means we aren’t direct descendants in exactly the same line - 13 cousin levels ago, our ancestors were siblings and then went down two separate lines.
That doesn’t make sense. Adjusting your number 13, what you seem to be saying is the equivalent of:
…my first cousin once removed. The once removed means we aren’t direct descendants in exactly the same line: one level ago, our parents were siblings and then went down two separate lines.
Wouldn’t that just be 13th cousins, in your case? You seem to be using “removed” to mean “cousin,” or I’m seriously misunderstanding.
Yes, I meant grandparents. I knew there was something wrong with the way I’d phrased it but couldn’t figure out what, so decided to post because someone else would spot it.
But what if you’re from Oklahoma?
You wouldn’t be “in exactly the same line” unless one of you were the other’s parent, grandparent, etc. Even without the “once removed”, you still wouldn’t be in the same line. The “once removed” just means that you’re in different generations, to the extent that even matters after that many generations. At least, from that specific ancestor: By the time you get that far back, it’s fairly likely that you’re related through multiple connections.
Why wouldn’t we be in the same line? That’s the usage of that phrase that I’ve seen and heard.
I had a friend who referred to her ex-spouse as her “first husband, once removed.”
Clearly, Randall Munroe is reading this board for comic ideas: