I'm not thinking of "current" DB engines, but an entirely new stack that is end-to-end JavaScript (or WASM).

Something like Java's Akka or .NET Orleans combined with React.

So the "data" would be persisted by stateful Actors, which can run either on the server or in the browser, using the exact same code. Actors running in the browser can persist to localStorage and on the server the actors can persist to blob storage or whatever.

Actors have unique addresses that can be used to activate them. In this system these become standard HTTP URIs so that there is a uniform calling convention.