Is it a bad idea to mention informal PhD offers in a job application?
tl;dr: No, you should not mention this "offer."

I wouldn't even call this an informal offer. An "informal offer" would be if you applied, and then were told by e.g. email or phone that you are admitted, and that the formal letter of admission will follow shortly. Encouragement to apply, no matter how strong, is not an offer of admission, and doesn't indicate that you would definitely get an offer of admission if you would have applied.

For example, you may sound great according to whatever information you sent to that professor, but your full application may reveal other things that lead the admissions committee to reject your application. Or maybe you really are that great and the professor has funding, but another, better, student applies that year and gets the offer instead of you.

Furthermore, individual professors have varying ability to actually influence the admissions process, depending on how graduate admissions actually works in that department.

Even for a "real" offer of admission, it wouldn't necessarily be a good idea to list it on a résumé. See for example this related question on listing fellowships that are offered, but which you decline. But being encouraged very strongly to apply to something is certainly not a CV-worthy accomplishment.