Do I need different linked server remote users?
The technical answer

To my knowledge, there's no problem in using the same user for all "client" machines. The same user account can connect to a SQL Server as many times as the licensing of that SQL Server allows.

The security perspective

However, you may want to review the security aspects of this setup - do all the remote machines have the same access requirements, or should they have different users for security reasons?

Could you perhaps even use windows/AD authentication? Remember that when you hard-code an account into the linked server, pretty much anyone who can log on to the remote server will gain the privileges on the "central" server that you've given to the user account defined on the linked server, which may lead to a privilege escalation.

Tread carefully with linked servers and permissions, is all I'm saying.