I just tried it on a clean install and the issue is there. 
- From a clean install with CB & CB Subs I installed sample data. (Joomla 4.4)
- Added a new Plan in CB Subs
- I created another Template style from Cassiopeia, made some changes and assigned it to the Home page (maybe don't even need to do this as the default data loaded assigns modules to the home page)
- The original template style remained as the default template.
- Then on the front end I log in and go to my profile page. 
- In the subscriptions tab, I select the plan and click on "upgrade" 

Here's where the template changes to the home template, plus all modules assigned to the home page load here as well. 

I set the "Search Engine Friendly URLs" to NO in order to follow the itemid value, and that's when the itemid goes missing from the uri. 
In Joomla 3 the menu would default to the template marked as "default" while now, if the itemid is missing, it defaults to the template of the home page with all the modules of the home page (home page itemid??)

I can assign "hidden" menus to those pages and that solves the issues for CB Subs. I have to create for all of them, if not, the ones I don't create a menu for, default back to the home page template.

It doesn't happen with the PMS, that retains the itemid OK. 

I noticed that if I remove the itemid from the url in any other place, like viewing an article, then the template changes to the home page template and not the default template. So this is why I'm saying it's a J4 issue, a missing itemid results in home page template and modules. 

It may be a purposeful decision, but it means that any errors are highlighted big time. 

I'm not sure that there's anything you can do about this other than check how to keep that itemid in CB Subs plans. 

Thank you,
Tomás 
 

Thank you!