The collective wisdom seems to be to leave it alone, so I will take that path. Thanks to everyone for helping me puzzle this out!
Just to make sure, she calls him "Papi" in Spanish, right? She doesn't go around calling him "Daddy" in English, and only translated it as an explanation to you? If she's doing the former, that's cool. As noted, for the most part, people who know enough Spanish to understand will also know it's a normal term of endearment. If she's saying it in English, I'd mention it to her.
I lived in Italy for several years and had numerous embarrassing moments because I knew the literal translation of a word but wasn't familiar with the cultural nuances tied to it.
I once tried to express to a date that I felt at ease in his presence. My Italian was still pretty iffy at the time, so to clarify, he asked, "So...like an armchair?" I responded, "Exactly! Like a comfy, warm armchair." and didn't understand why he seemed displeased. He didn't call again, and it was only later that I learned that calling someone an "armchair" in Italian (
poltrona) was roughly equivalent to calling them a doormat in English (although not exactly. It's more like you're saying you're settling).
I also discovered the hard way that a joking "You're so full of crap" (in the jocular sense it's used among friends in the US when someone tells an obviously overblown tale like "I went to school with Bigfoot!") didn't have an Italian equivalent. I tried to translate it directly and... well, I basically told my friend he was a piece of shit. He was understandably angry at the insult out of nowhere, and even after I apologized and tried to explain, our friendship was never quite the same.
I still cringe at the time I was invited to a dinner party, back when my Italian was still so poor that I mostly just listened to the conversation to see if I could at least pick up the occasional word or phrase. I heard one and proudly told my hostess in my broken Italian, "Hey, I know that word.
Trombare. It means to play the trumpet (
tromba), right? Everyone heard me and busted up laughing. Technically,
trombare does mean to play the trumpet. Unfortunately, it's much more commonly used as slang for... well, let's just say that someone making an Italian porn parody of E.B. White's
The Trumpet of the Swan wouldn't have to change the title.
In short, translation errors will happen when you're learning a new language, and those nuances can be the hardest part to pick up. Translating dictionaries generally won't explain them. Really the best source of information like that are friends and acquaintences.