How to switch to other site's database in multisite enviroment?
You can't.

Multi-site is only about sharing the code in the file system. Unless you actually add the database connections to the specific settings.php, Drupal doesn't even know that there are other sites.

Functionally, once the bootstrap and site selection happened, there is 0 difference between two different installations and a multi-site.

IMHO, there are more downsides than advantages with multi-sites these days, you can use git and e.g. different branches to share code between multiple sites at least as well, with way more flexibility and options for updating and customizing them.