If this isn't your neighbor's first garden, I'd assume they're correct, cucumbers can definitely grow like this. There are even heirloom varieties that always grow round, they're also more yellow than common cucumber varieties. Easiest way to tell would be to cut into one.

I grow lemon cucumbers that look pretty much exactly like that, only difference is there’s slightly more pronounced striping.

Cut it open, smell it, report back.

Probably a lemon cucumber

I grew some Indian cucumbers this year (in the UK) called 'poona kheera', they started out almost white then turned orange when starting to ripen! They weren't quite as spherical as yours but not by too much. They tasted like regular cucumbers but were particularly crunchy.

You should be able to make recipes that normally use cucumber, they should taste pretty normal. It may be due to the strain of plant, but this can also happen to normal cucumber planta if the roots are stunted and they don't get enough sun. Color-wise, I have a few cukes that look like that from my garden, and the round shape just betrays the fact that cucumbers are actually a kind of melon!

cucumbers are just weird pumpkins/gourds, so yeah that's a "cucumber".