This exists, its called a user. You just use different database users for this.

I was thinking of something that only exists at the pool level.

Every new version of your app has the potential to change behavior in a way that would affect the previous version if the connection was recycled during a progressive rollout.

But I don’t think I would want to create a real database user for every version of the app.

I suppose the connection pooler could map versioned users to the same real user, and use separate pools, but a dedicated UA field is probably better.