Can you link 2 different mysql servers to read and write from each other?

This is an example of distributing databases and having them synchronise. It's a common practise for applications with enormous user bases (ie. YouTube) to optimise for speed, by having multiple databases available for reads and writes, and having distributed systems synchronise the data when resources are available. Interestingly, if you ever refresh a YouTube page and notice different view counts, it's because it's probably hit a different server which has a different, cached version of the view count, so that number can go up and down!

In theory, the priority here would be that your website needs to perform immediate operations on the primary database, while your website database can operate off some slightly stale data. For example, if you close your account on the website then you should not be able to login to the game effective immediately.

In this situation I would recommend having the website have an available connection pool to your primary database for important write operations (purchases, account changes). For anything else, I'd recommend having a scheduled task in the background perform any synchronisations which might be needed. For example, updating the website with acquired trophies or achievements.

This would also mean your Unity server should never connect to the website database, have the website responsible for managing any synchronisation and alleviate this responsibility from your game server, freeing up more resources for a better gameplay experience.