Isn't DAX only a network-based in-memory cache, and not a in-process in-memory cache? Or is it both?

It's definitely at least network-based, b/c you have to spin up instances of it in your VPC.

However, they also require custom clients, which is a little odd afaiu (b/c in theory DAX is API-compatible with DynamoDB, so why would regular dynamodb sdk clients not work?) so maybe the custom clients also add some in-process-based caching?

Funny enough, at my company we did just this. Having local state is a requirement for our uptime and reliability goals, and we have a dumb service that pushes this back to the services via DynamoDB streams, and also dumps the data into S3 in case that service is down.