All Joomla modules can support a surrounding CSS class. That functionality is just built into Joomla modules. Edit your module then under Advanced add whatever CSS classes you like to "Module Class". Then you can supply whatever custom CSS you like to CBs template overrides functionality below to have your custom CSS output automatically handled.

www.joomlapolis.com/blog/kyle/18711-template-css-overrides-made-easy

Ah, I see. Will implement that for next build release so it's easier to target styling the count specifically.

Is it possible to overwrite it somewhere without losing it after the update?

It's possible to override templates, yes. You however will be responsible for maintaining any overrides as differences may arise in future releases. To do this create a new folder in the following location.

/components/com_comprofiler/plugin/user/plug_cbactivity/templates

The folder name will be your new template. So for example the following.

/components/com_comprofiler/plugin/user/plug_cbactivity/templates/custom

Next copy over whatever template file you're wanting to override. In this case it's the following.

/notifications/button.php

So now you should have the following.

/components/com_comprofiler/plugin/user/plug_cbactivity/templates/custom/notifications/button.php

Finally within CB Activity > Parameters > General set Template to "custom". Now you can edit button.php however you like and it will be applied to all notification buttons.

Latest CB Activity build release now includes classes for the notification button icon and count.