Can Site studio templates be saved into the database?

Yes, the generated templates can be saved to the database.

As of version 6.3.5 the template storage can be switched from the file system to the key value storage. If enabled Site Studio will be using the database more often as it will save all config templates in the database. 

To enable this feature you need to change the alias of the cohesion.template_storage services to point to cohesion.template_storage.key_value.

You can do so by providing a site specific services.yml and provide the path to it in your settings.php

for example in your env.services.yml:

services:
      cohesion.template_storage:
             alias: cohesion.template_storage.key_value

Then in your settings.php:

$settings['container_yamls'][] = '/path/to/env.services.yml'

Acquia Site Factory

This feature is especially useful when utilising a distributed filesystem as found on Acquia Site Factory, however, the above approach applies this change throughout your platform. 

For a more flexible per-site approach you should consider installing the Site Studio ACSF module