"Tety" is nominative plural here, right? Then why can't the sentence be translated as "We can see their aunts"?
I'm confused about the "svoje". Because in a different context it can be translated as "their" or "my", but apparently not here.
"Svoje" can't be directly translated to English, there no such word there. It always refers to the subject of the sentence. And because the subject here is "we" it must be "our".
So "Vidíme naše tety" would be incorrect if we follow this logic? Even if it would work in Germanic languages...
It is possible, mainly to stress that they are really our and not someone elses.
And, I missed it, it must be "We see OUR aunts.", not your. svoje refers to the subject.
It can, it is accepted without the can.
Can see is just commonly use instead of the plain see in English. See https://en.m.wikipedia.org/wiki/Uses_of_English_verb_forms#Have_got_and_can_see