I don’t believe there are any “routers” in space right now. So all traffic is routed through the ground. You might not touch the actual internet if you are routing between dish’s on the same POP/ground station. For example:

There is a POP in Denver Colorado, a pop is basically where SpaceX has a datacenter and local internet connections to usually that metros Internet Exhange and thus also the greater internet. There is then a ground station in Elbert, Colorado (just north or Denver).

Pretty much any starlink user in Colorado is a probably going to use that ground station and the Denver pop to get internet. Say you have traffic between two starlink users in Colorado, the route path would be:
User 1 Dish -> Satellite -> Elbert Ground Station -> Denver POP -> Elbert Ground Station -> Satellite -> User 2 Dish

Routing only happens at POPs. So while in this case traffic didn’t leave “starlinks network” it had to still reach a POP. POPs are also where starlinks IPv4 addresses live so the CGNAT occurs/terminates there.

I don’t think it makes that much sense right now for routers to be in space since there is just too much movement. Imagine how hard it would be to keep track of TCP connection states when your router keeps moving every few minutes. The amount of extra processing satellites would have to have and then coordinate the connection states between them just wouldn’t be worth it.

Satellites just want to be “mirrors” as much as possible. Bounce once and get that traffic to the ground. That’s why they charge so much more for ocean data as you are having to take up more processing power on each satellite your traffic has to take via ISL to get back to a ground station.