I guess it's japanese thing how you called someone with chan = close.I don't get what happened in that scene where he asked her if she was close to Mai. What did her saying Mai-chan mean for him to ask that?
then it's flashback to how everyone even her family called her just mai. except her best friend,yukari. who kinda win him in term of being closer to mai(the part where they stay at grandpa house then she scolded him for calling yukari weird)
So in kurobe mind might be like, Do you think you're that close to mai? like yukari who won me? something like that,I guess.
With him being that "jealous" then probably give her death stare enough to make that girl run away.