this is /apni/ whereas you are thinking of /aap-ki/. See the difference? apna/apne/apni means "one's own/ self's". There's a grammar rule that says one must use this word instead of my/your/his/her/our if the thing is being possessed by the subject of the sentence. You can say "I am with his mom" or "He is with my mom", but you can't say "he is with his mom" or "I am with my mom" :) You gotta say, "I am with one's own [aapni] mom" or "He is with one's own mom".
Why is "I am with her mother" marked as incorrect? How do you know it's "my mother" and not "her mother"?
अपना/अपनी always refers to the subject of the sentence (or the subject of the clause that अपना/अपनी is in if the sentence has multiple clauses). Since the subject of the given sentence is 'I', अपनी translates to 'my'.
If the sentence was 'वह अपनी माँ के साथ है' instead, then the subject of the sentence is वह(he/she). So अपनी is 'him/her' with the sentence translating to 'She is with her (own) mother'.
If you want to write 'I am with her mother', you have to use the actual word for his/her (उसकी ). 'मैं उसकी माँ के साथ हूँ'.