I have no idea why a toddler would be unhappy about having a baby stare at them.
I do know that for your toddler, their life has been turned up side down with the arrival of baby. Your toddler had the adults all to themselves, but now they have to share, and 2 year olds are not known for being good at sharing.
As for what to do, often we can redirect a young child by making something into a game. For example, racing a child to their bed when it's bedtime, where we sometimes win and other times we let them win, and this makes it a game.
Many families play games about staring at each other, so it might help to introduce a staring game to see if it helps your toddler focus on the game instead of their dislike for the baby staring.