what are good database applications for web design with os that will always be compatible?

That’s a pretty big question. A really big question. Is this for client, or server, etc? Writing an app, or creating a service and its server, or both? Things to use to learn, or things used to deploy? What is your initial scale?

SQLite is portable ~everywhere, fast, robust, but not usually directly exposed to the web.

Filemaker Pro is an excellent database and platform for app development on macOS, and offers web capabilities.

For database portability and general database capabilities, PostgreSQL.

If you’re just getting started and looking for some development-focused tooling that is packaged for ease of installation and usage and this particularly for learning some of the common tools, MAMP might be an option.

For a platform intended to dela with a whole lot of client and server app development, Parse. For a “simpler” (but capable) backend, OwnCloud.

It’s not all that common to be accessing an SQL or other database directly from JavaScript or Python or something else compiled into JavaScript or WebAssembly or whatever else, without having that particular language’s frameworks involved, or the rest of the chosen platform involved. But I’m not sure of even which end of the usual connection you’re working with here.

If you’re running your own public web servers and database servers for whatever this is, you just acquired some non-trivial networking and particularly security requirements, too.

And there are trade-offs here. Everywhere. And the more portable, the more problematic things usually get.

PS: for finding more stuff from the above examples, the alternativeto.net service.