I received [an A / As] on my exams

Solution 1:

I feel exactly the same as you when reading both sentences together. However, we mustn't let ourselves get carried away by grammatical conundrums. The logical meaning of both of these sentences is that you got an A on each of the exams you did, as it's not possible to get a single A for multiple exams, nor various As for one test. I believe anyone would interpret it more simply if they didn't pay close attention to the grammar.

Solution 2:

I agree with KSE Academy's answer: in practice, neither sentence implies anything incorrect, since contextual knowledge will disambiguate the meaning.

That said, the second does sound better to me:

I received As on all of my exams.

If you wanted to be more "precise", it is possible to say something like

I received an A on each of my exams.

but as mentioned, there's no practical reason to be this precise with this sentence.