The blue ring around your space isn't a diplomatic line. It is a cultural presence line. It shows how far your culture extends. Most people treat it as a diplomatic line, but the AI does not. There was a discussion about it in the latest dev stream. There is also an open borders treaty already. Possibly not the response you were hoping for but I hope it's helpful.

I think, in the future (right now it's too late), we should do something about that.  Open borders doesn't mean anything right now.  And since we only see the influence line, I don't really know if the AI is doing stuff in my diplomatic lines or influence lines to try and counter my influence.

If we were to forbid travel inside another nation's influence zone as it is now, many planets would be stranded behind ennemy lines, both mine and the AI's.  It's not uncommon to see a large section of a nation's empire be in another nation's influence zone.