You can think of 'different people' as different teams, hence different Teams. This is just how Teams was designed. I'm not sure how you can customise Channel membership without going down the Private Channels route (Note: never use this personally, don't have much experience).
You might want to consider a 'department' Team site e.g. 'Projects', with a SharePoint page as the Team's 'homepage'*. That page could show all your relevant Projects / Teams. Note: each Team a user has access to would still be listed in their Teams list but might be more welcoming to your users.
*I can't remember how to do this but you can basically set a SharePoint site as the first thing seen when a 'department' Team is clicked, like a department Intranet page.
RE notifications ... yes, each Team and Channel has a default set of notification options, though users can still modify these defaults to suit their preferences. Note: these defaults are set across all Team members. Not sure how this affects Private Channels.
If you are left with your only option being multiple Channels for multiple projects i.e. with the same membership, you can use Team Tags to <at>mention individual or groups of Team members e.g. per project, but this sounds like a right faff to setup and maintain.
When creating a new Team, you do not have to post about it; there's a checkbox somewhere in the Create Team wizard. No post, no notification to members.
Good luck.