It is "his" home... he sees the other children as invading his space. 3 years is a whole other developmental phase.
He can't always be expected to "share" everything... nor will a 3 year old have COMPLETE control over their feelings, nor the coping skills for it, nor the ability to "know" what feelings he has or why.
I babysat too... and had kids in my home. My daughter, with this one child, had a hard time. ANd my daughter generally does not have a problem with other kids. The other child was REAL controlling and bossy. I eventually had to not babysit that child. It was a personality conflict. I also DID NOT make my girl "have to" share everything, and certain rooms were ONLY for her... where she could go to do what SHE wanted, alone or otherwise.
Its not so much a "fault" of your son... he is behaving developmentally for what he is "capable" of doing. Many times, when a child is expected to "be" something that is not what they "can" do yet or understand... they will act out. Frustrations. The emotional maturity... is still emerging.
He does not feel understood nor validated for what he is going through nor his feelings.
LET him have things and places in the home that is JUST HIS... and that he does not have to share. If there are times he does NOT want to socialize, so be it. If he NEEDS his own space and his own time alone... let him. Because once he starts acting out... this means he is PAST his threshold for tolerating it.
Tell him you understand... he needs to hear that and know it. LET HIM have his own "boundaries" about his comfort level.
And, he could also be just over-stimulated with all the other kids around, and over-tired. ALLOW him to have a breather sometimes. It is "his" home too.
Even adults, get "tired" of being social too sometimes... and just need to have time alone. At their own pace and NEED to re-coup and have respite.
All the best,
Susan