There are a lot of reasons that you might want Postgres in the browser, but one of my favorite possibilities is to use it as a local data store (similar to mobx or redux)

The Electric team have built this extension which will fit that use-case very well: https://pglite.dev/extensions/#live

Playgrounds like this which don't need server resources.

Adding offline support (PWA) to an application which was primarily designed for use on a server. Though I'm not sure how well that'll work in practice, since the rest of the application will be designed to run on a server as well.