'A pair of shoes is/are odd.' Is there a name for this double meaning regarding singularity/plurality?

To some extent, OP's example "puns" on the overlapping senses of a pair = one set OR two items and odd = not even (of integers, not a multiple of 2) OR unusual...

pun Cambridge Dictionary
a humorous use of a word or phrase that has several meanings or that sounds like another word

...but I don't think it's quite a "pun". It's just...

wordplay
the activity of joking about the meanings of words, especially in an intelligent way

As a general principle, there shouldn't be any difference between OP's "attributive adjective" usage (an odd pair) and the "predicative adjective" version (a pair that are odd).

But because the attributive usage is more common in this exact context (for the "non-contrived" sense of an unusual pair of shows), I think the "wordplay" sense comes across better with the predicative format (we're naturally inclined to seek an unusual meaning when presented with even slightly unusual phrasing).

Here's a link to an NGram usage chart showing that pair of shoes is is 4-5 times more common than pair of shoes are, but I really don't think that "stylistic choice" makes any difference to OP's example of wordplay.