Counting each filter near its label I've created a view in order to override the default search page form and results, then I've exposed the content type in order to have the possibility to filter by that, all now is working

How can I have the count of the relative content near each content type label on the filters?
Sounds like you want to use search facets.  Facets are ways of showing various search 'tag' information for the current results with/without counts.  Note that this is slightly different from a filter which will show all values even if there is no result with the specified search criteria.

That said, there is a defunct facets core search module. As the module page says, the best way to approach this with Drupal 8 is to use the Facets module which requires the Search API module with the included Search API DB engine (no Solr required).

This will add a bunch of extra functionality that the core search will not handle.  Stuff like full text search across multiple text fields, stemmed searches (e.g. search for work also finds working, works, and the like), ability to show extracts (were the word was found), and much more.