You know this name was a human name way before some people decided to make it a name for their dogs - but somehow I don't think French people or those from francophones will have that strong a connection. Though I could be wrong. However I think it is very cute as a nickname.
Utterly silly and immature, even as a nickname. There is no way any woman named Fifi could be taken seriously. People would think she's ditzy simply for not changing her ridiculous name.
My daughter is named Philippa, and somehow she became known as Fifi. I'm not sure how this came about. It wasn't planned this way. I think I started calling her "Fifi" affectionately when she was very small, because neither I nor my husband liked the nickname "Phil". Fifi (my 16 year old) doesn't mind the name, and actually prefers it to "Philippa", which I think is beautiful. Objectively, I agree, though, that the name sounds cutesy and infantile. But it's not strictly a name for animals!
My niece, 3 years old, is named Sophia. She gave herself the nickname Fifi when she started talking. At first I thought it sounded a little silly and funny, but especially since my daughter, also 3, took right to calling her that, we do all the time, so it sounds normal to me now.
When I was a little girl, I had a neighborhood friend whose nickname was Fifi. (The Lord only knows why, since her given name was Elizabeth.) Despite this pleasant early association, I can only see Fifi now as a name for a pet, a drag queen, or a sex worker. (Or maybe a clown?)
Fifi has always been a stereotypical French-sounding name, one of the most famous "Fifis" I know is Fifi LaFume, the young female skunk from Tiny Toon Adventures.
Personally, I have always viewed "Fifi" as a direct insult to one's intelligence (whether or not it was originally intended); terribly undermining. It receives the same ratings as gifting your child the name (or nickname) of "Froufrou," from me.
That being said... I have a cousin by the name of "Sophia" who answers to it (as she cannot stand "Sophie.") Various family members and friends summon her by it, but I stand firmly in using her actual forename.