You con't say how old the child is, but the part where you asked if he'd treat his other grandmother this way jumped out at me.
Why would you ask that? His other grandmother is who she is, you are who you are. Nobody, not even children, are obligated to feel the same way about everyone.
Respect his bodily autonomy. Don't try and push for him to sit beside you, hold your hand, etc. All that's going to do is increase his discomfort and reluctance.
From you OP, it sounds like you have some high expectations of your grandson in terms of emotional bond/relationship. Both of those things grow and evolve, and can't be demanded. The more you push, the less you're likely to achieve.