Our company has a domain MAIN_DOMAIN and I log into my Windows laptop as MAIN_DOMAIN\my.name. When I run SSMS on my laptop I can connect to both Server1 and Server2 using Windows authentication.

This is how the linked server is configured on Server1:

On Server1 the MSSQLSERVER service (sqlservr.exe) runs under user called NT Service\MSSQLSERVER.

On Server2 the MSSQLSERVER service (sqlservr.exe) runs under user called NT Service\MSSQLSERVER.

A different user can run the query involving linked server without problems.

He logs into Server1 using Remote Desktop and using other_domain\his.name. He runs SSMS on that Server1 and connects to SQL Server using Windows authentication.

His login on Server1 has all permissions as well:

Our company has a domain MAIN_DOMAIN and I log into my Windows laptop as MAIN_DOMAIN\my.name. When I run SSMS on my laptop I can connect to both Server1 and Server2 using Windows authentication.

This is how the linked server is configured on Server1:

On Server1 the MSSQLSERVER service (sqlservr.exe) runs under user called NT Service\MSSQLSERVER.

On Server2 the MSSQLSERVER service (sqlservr.exe) runs under user called NT Service\MSSQLSERVER.

Here is output of setspn -l. I get the same output when I run it on server1 or server2:

C:\Windows\system32>setspn -l SERVER2
Registered ServicePrincipalNames for CN=SERVER2,OU=Azure Resources,OU=Corporate,DC=other_domain,DC=com,DC=au:
        MSSQLSvc/SERVER2.other_domain.com.au:1433
        MSSQLSvc/SERVER2.other_domain.com.au
        WSMAN/SERVER2
        WSMAN/SERVER2.other_domain.com.au
        TERMSRV/SERVER2
        TERMSRV/SERVER2.other_domain.com.au
        RestrictedKrbHost/SERVER2
        HOST/SERVER2
        RestrictedKrbHost/SERVER2.other_domain.com.au
        HOST/SERVER2.other_domain.com.au

C:\Windows\system32>setspn -l SERVER1
Registered ServicePrincipalNames for CN=SERVER1,OU=Azure Resources,OU=Corporate,DC=other_domain,DC=com,DC=au:
        MSSQLSvc/SERVER1.other_domain.com.au:1433
        MSSQLSvc/SERVER1.other_domain.com.au
        Microsoft Virtual Console Service/SERVER1.other_domain.com.au
        Microsoft Virtual Console Service/SERVER1
        Microsoft Virtual System Migration Service/SERVER1.other_domain.com.au
        Microsoft Virtual System Migration Service/SERVER1
        Hyper-V Replica Service/SERVER1.other_domain.com.au
        Hyper-V Replica Service/SERVER1
        WSMAN/SERVER1
        WSMAN/SERVER1.other_domain.com.au
        TERMSRV/SERVER1.other_domain.com.au
        TERMSRV/SERVER1
        RestrictedKrbHost/SERVER1
        HOST/SERVER1
        RestrictedKrbHost/SERVER1.other_domain.com.au
        HOST/SERVER1.other_domain.com.au

Unfortunately, I don't understand what it means. The only thing that I understand here is that MAIN_DOMAIN is not mentioned anywhere.

A different user can run the query involving linked server without problems.

He logs into Server1 using Remote Desktop and using other_domain\his.name. He runs SSMS on that Server1 and connects to SQL Server using Windows authentication.

His login on Server1 has all permissions as well:

But, there is another user who can run this query with problems.

But, there is another user who can run this query without problems.