No, just access rights that may have impact on visibility 😉

May I know the details? because here is the problem that the same profiles sometimes can not search and others with the same profile can. What setting determines this?

Allocation to projects.
By defaut, users with non adminsitrator rights have access only to items of projects they are allocated to.

This is an example of a profile with high privileges, it is still not clear to me. The user enters the activities tab, or projects and they can search by name only what they are allocated to? What about the situation where they have allocations and it doesn't search for anything by name.

The user enters the activities tab, or projects and they can search by name only what they are allocated to?

Yes, if his profile gives him read rights to items of his projects

What about the situation where they have allocations and it doesn't search for anything by name.

Same thing, same restriction : they only see what they are allowed to see