Well, one might say it in certain contexts, "I, being a teacher, am in a parent-teacher meeting presently. I'm here with a mother". But yes, in most contexts, speaking of one's own mother, it'd be "I'm here with mother/mum/mom/etc", or else "I'm here with my mother/mum/mom/etc", to be clear.
This course teaches Standard Ukrainian. Standard Ukrainian is based on the Kiev speech. In Standard Ukrainian, «є» is not normally used in this sentence.
Different dialects use «е» differently. In some dialects, you would use «Я тут є з мамою». This is not literary Ukrainian, and this would sound strange in Kiev.
Looks like your native dialect is different from Kiev speech in this regard. It's great that you speak such a dialect. Treasure it, because it's an important heritage. However, this course does not teach your dialect, it teaches literary Ukrainian. The language used for literature, news sites and documents. In literary language, this sentence doesn't require «є».
Just remember that Ukrainian is not a single form, it has different variants. This course teaches one variant (literary Ukrainian based on Central Ukrainian dialects), your native variant is different (likely some Western Ukrainian dialect). Not everything that is correct in your native dialect works in literary language.