wrong indication milage equipment

For hiking, I also use the running profile (without HRM).
Once the transfer for this activity is done to GC, I change the type of activity from running to hiking in GC.

But since I added my new shoes, the distance of the hiking (when importing to GC it is still a "running" activity) is added to the milage from the shoes.

Rule-based automatic assignment of gear to an activity is done upon upload/import. When you sync your watch and upload a ‘hike’ activity actually of [FONT=Courier New]Type[/FONT] Running to Garmin Connect, GC will scan your Gear data table for any items to be assigned as default gear for Running activities (because you asked it to by way of the gear item's user-configurable attributes), and create link records between the new Running activity and any such items it finds.

After I changed the type of activity, the distance from the hiking activity is still counted in the milage from the shoes (so slightly incorrect because I don't hike with my running shoes).

Manually changing an activity's [FONT=Courier New]Type[/FONT] (say, to Hiking) after that will not cause any default gear assignment rules to be checked again, or any link records to either be added or deleted automatically.

Every time you go to the Gear pages in Garmin Connect online, the link records are used to identify the activities of which the distance is to be added to each gear item's total mileage that GC calculates just-in-time with no caching.

So to solve the problem, I disactivated the automatic count for the shoes and the running activities.
Now I have to add the gear manually to each running activity.

You cannot ask Garmin Connect to assign your running shoes by default to only some activities recorded as of [FONT=Courier New]Type[/FONT] Running but not others.

Is there a possibility to change this…

Yes, of course. You can resume asking Garmin Connect to assign your running shoes to all new activities of [FONT=Courier New]Type[/FONT] Running by default, and then manually removing your running shoes from the gear linked to your Hiking activities while you're editing those activities to change their [FONT=Courier New]Type[/FONT] from Running. Either way there is manual effort involved, but you can change what you have to do procedurally.

… or do I have to add a new activity profile to my watch (hiking) ?

Only if you can create a new profile that will record activities as being of [FONT=Courier New]Type[/FONT] Hiking, irrespective of the profile name you end up giving it. If Hiking is not a sport you can select for the new profile, then it won't work how you probably imagine it would.

I'm not a fan of adding other profiles, watch screens and other stuff to my watch if i'm not sure it's not going to mess up my watch.

Fair enough. Are you a fan of manually editing gear for activities after upload?