Yes, but each database instance has a limited number of connections that can be open. You can use Lambda to handle as many web requests as you want, but if each Lambda invocation is creating a new database connection you've only shifted your bottleneck one layer down the stack.

I think Azure runs pgbouncer for that.

Sure, you could run pgbouncer yourself on EC2, but unless it's offered as a managed service you're no longer "serverless" as you've got a pgbouncer server to administer yourself.

I agree. That's why Azure runs it for you.