I actually find it quite insulting. I worked incredibly hard to become a teacher. A degree, followed by a PGCE and then a Master's. It also cost me a shit load of money. I've had to carry out numerous research studies, lots of teaching practice and external courses, plus all the other things TAs don't have to do. EHCPs, parents evening, reports, tracking attainment, end of key stage assessments, book scrutinies, lesson planning, differentiation, marking, spending a load of my income on resources, working a 60+ hour week. The list goes on.
A TA is not a teacher, in the same way that a flight attendant is not a pilot. The roles, the training and the knowledge required are vastly different, and I've done both.
My step sister is a TA. She has very low levels of literacy and numeracy, her speaking is embarrassing (things like dropping h's, leaving t's off the ends of words, saying th as f, mispronounciation, inappropriate slang, absolutely no general knowledge etc. She tells people she's a teacher. She couldn't qualify as a teacher if her life depended on it. People mock her behind her back because it's blatantly obvious she's not a teacher. (She's an awful person, so attracts negative attention).
If someone who is a qualified teacher changes role to a TA, they're still a qualified teacher and that's fine. A TA who has no qualifications, or a level 1/2/3 in being a teaching assistant, then they absolutely should not be calling themselves a teacher.
As for people saying it's "snobby" to think that way, it's just another example of how English people sneer at academia and education. I wonder how you'd feel about a nurse calling themselves a surgeon and attempting to perform a heart bypass on you?