Some pets seem to always have the option to research all or even one gene, grayed out. Despite the fact that I have enough vermillion and silver on hand.  It seems to be the pets I have gotten out of the pound, more so than the wild caught ones.
Though not all my pound pets seem to forbid me from checking their genes.

Am I missing something?
Like if they are on an island is that the reason I cannot check their genes? Or is this some kind of, glitch/bug?

Indeed, if they are exploring or otherwise busy, you cannot research their genes. This includes any and all activies a pet can be involved in, like being pregnant, being on a quest or crafting.