The best AWS Documentation implies - YES, it does work
Specifically for Aurora Serverless, there is an advisory that "LISTEN/NOTIFY" is NOT RECOMMENDED. This implies that it's possible with Aurora Serverless but not advisable. It also implies that normal RDS LISTEN/NOTIFY does work and is ok to use.
https://docs.aws.amazon.com/AmazonRDS/latest/AuroraUserGuide/aurora-serverless.html
Currently, certain SQL queries and commands, such as cursors with hold
(DECLARE name ... CURSOR WITH HOLD FOR query); session-level advisory
locks; temporary relations; and asynchronous notifications (LISTEN,
NOTIFY commands) prevent scaling and are not recommended
My own quick test on AWS shows - Yes, it does work
- Launch Instance with Public Access
...
- Allow Traffic
- Run command in dbeaver
Note: This test doesn't show a successful receipt of a notification. That will need a quick console application to test that.