It’s definitely the same thing. We can test this using other modals and auxiliaries in equivalent question constructions to show that we’re dealing with analogous structures:

If making a question with “might”, for example (with the pro-predicate base sentence “But your medical clinic might do”), we get “But might your medical clinic do?”

With “would”, “But would your medical clinic do?”

So, with “dummy do”/do-support leading to the insertion of “do” for inversion purposes, along with the separate pro-predicate “do” lower in the clause, “Your medical clinic does” (or possibly “Your medical clinic does do”) becomes in the same way “Does your medical clinic do?”

Here’s a blog post about it, and here’s a StackExchange exchange about it.

This is common in British English.

For example, the question “Are you going into town?” might be answered by an American with, “I might,” and by a Brit with “I might do”. In past tense it would be “I might have” vs. “I might have done”.

This is all perfectly systematic and grammatical - this person just has a different grammar than you do. Though I guess that’s what Nazis do best: enforcing arbitrary standards in systems they don’t understand to destroy diversity to everyone’s detriment.