"Tag" is a game. It's a really fun game.
A "tag" is "a game played by two or more children in which one child chases the others and tries to touch one of them. This child then becomes the one who does the chasing."
In this game, often there is a pole or something that is chosen to be the safe place. When the person being chased runs to this safe pole and touches it, this person cannot be tagged (while they are touching it).
"Players may be safe from being tagged under certain circumstances: if they are within a pre-determined area, off the ground, or when touching a particular structure." - Wikipedia.
This happened when Snape said to Lily that she was a witch. Lily marched towards Petunia who had already ran to the swings. They were both nervous and scared of Snape. "He flapped after the girls, looking ludicrously batlike". The sisters were holding on to the swing pole as if it would keep them safe.
This could have been an instinctual reaction. Perhaps they played "tag" a lot, and their immediate response to Snape chasing them was to get to the safe place where no one was allowed to touch them. Also, when frightened, we love to hold on to firm structures or take shelter behind trees.
Here is how adults play [the new] tag. There are no safe areas or safe structures. There is a time limit though. And it's one on one. It is a hybrid of Tag and Parkour.
Source: World Chase Tag Tournaments.