"We got her!"
"ją" is Accusative form of "ona". You use "mieć" with Accusative so "(My) mamy ją" (We got her).
"jej" is Genitive form of "ona". Accusative in negative sentences switches to Genitive so you would say "Nie mamy jej" (We didn't get her). Genitive is also used to express possession of something so "jej dom" means "her house".
If that's the case I'd say "we got her" is incorrect english? People say "I got this" or "we got you" to mean the present tense but that's understood to be slang, its most popular usage is in African American Vernacular English. Which is kind of controversial as to whether it's "correct" english, but it definitely doesn't have a place in an academic setting to teach language, especially if it were the only sentence in this course written with AAVE grammatical conventions.
The sentence "We got her", in english, unambiguously means "in the past, we got her." As in, we captured you, or we picked you up.
If you wanted it to mean "we are in possession of her in the present," or "we have her," you'd have to say it as "We've got her."
Which is just a contraction for "We have got her" anyway, in which case you could just drop the word "got" from the sentence and change it to "We have her" and not really affect the meaning of the sentence. So it seems like "We have her," which is the most direct translation of "Mamy jej," is a much better sentence to use.