Your screen is inconsistent. Below you have checked "Be made using the login's current security context", yet you have a password above. You don't have a password with self-mapping, since users log in with their own credentials.
Anyway, the next time this happens, run this query, once you have changed the password:
SELECT ll.*
FROM sys.linked_logins ll
JOIN sys.servers s ON ll.server_id = s.server_id
WHERE s.name = '<RemoteServer>'
Save a screenshot of the result set somewhere.
Next time when the linked server stops working, run the same query again before correcting the password. I would expect the column modify_date to have changed. That is, something or someone have changed the login.