Is the word "uncle-in-law" commonly used when talking about the husband married to your mother's sister?
Or is this person typically still called your "uncle", even though the person is not a blood relative?
No, not in my experience. My mother doesn't have a sister, but my father does, and his sister's husband has always been "Uncle Jim" to me. I would not occur to me to use the term "uncle-in-law" for him.
Indeed, I would assume it would refer to someone who is the uncle of my spouse. That is, it seems to me that, if we used such term, my spouse's parent's sibling or my spouse's parent's sibling's spouse would be my "uncle-in-law". But we really don't.
I have never heard "in-law" used with aunt or uncle.
When you say "commonly used" in your question, this is how I think of it.
As such we used "aunt" or "uncle" for (generally) older relatives that were not direct relatives (immediate family, cousin, nephew, neice etc.), especially if the relationship was somewhat distant.
Relation by marriage did not make any difference.
Of course, you could have an aunt or uncle (-in-law) that is younger than you. In this case you would call that person aunt or uncle, as a statement of the actual relationship.
When I was a youngster one side of my family was quite extended. "Aunt" and "Uncle" were used so much.
I've never heard Uncle-in-law used, although I would be able to figure out the relationship if I heard the term. In person, I would always call my mother's sister's husband Uncle. If I'm describing the relationship to an outsider, I would call him my mother's sister's husband, not my uncle-in-law. In-law is normally only used with brother-sister relations, or parent-child.
Also, I'd only use in-law with close family. So my brother's wife is my sister-in-law, but her sister is just my brother's sister-in-law, not mine.
According to Google Ngrams, 'uncle-in-law' is used about as much as 'aunt's husband'.
I occasionally use 'uncle-in-law'. My wife (who is Korean) and I (who am Australian) have been hosting her niece (who is Korean) while she studied at high school in Australia. When signing official documents together, I would add (aunt) after my wife's signature and printed name, and (uncle-in-law) after mine.
I also very occasionally use 'aunt-in-law' (I have never been as close to her as I was to my (deceased) uncle, 'niece-in-law' and 'nephew-in-law'. 'Cousin-in-law' also exists (and, is the most common of all these).
There is an ambiguity in all of these. 'Uncle-in-law' could be my aunt's husband (just as 'brother-in-law' could be my sister's husband) or my spouse's uncle (just as 'brother-in-law' could be my spouse's brother). In any case, we have the possibility of specifying: 'my mother's sister's husband' v 'my father's sister's husband' etc.
My mother's sister's late husband was my uncle.
In American English, the suffix "-in-law" is commonly used as part of these words:
Similarly, the prefix "step-" is commonly used as part of these words:
The terms "aunt", "uncle", and "cousin" are more general, and most Americans do not expect to know immediately the exact degree of kinship. If it is important to say, one could say something like "She is my wife's aunt" or "She is my father's brother's second wife" or "She is my wife's stepmother."
"Cousin" can include someone who is several degrees removed. For example, I can refer to my "third cousin once removed" as a "cousin".
I can even call someone who is probably distantly related to me (such as someone from my paternal grandmother's ancestral township who shares a family name) a "cousin" or "long-lost cousin". However, this can be taken too far, and risks being thought "presumptuous".
Indeed, the terms "aunt" and "uncle" are often used for people who are "family friends", and are not relatives. For example, suppose a family has a small child, and is on friendly terms with an (adult) neighbor. Suppose they let the neighbor baby-sit the child. The family might teach the child to call the neighbor "Uncle Jack" -- even though there is no relationship by blood or marriage -- to encourage the child to treat the neighbor with the same respect that the child treats his or her parents' siblings.
Before answering that, uncle is always used for people who are older than you. So your Father's brother is uncle as well as your Father's friend, who is not blood related is also called uncle.
uncle-in-law for a person married to mother's sister
Is technically correct but you will not find anyone using it. Since the meaning stays the same as simply uncle, people tend to use the latter only.