Your MSSQL server name will still be your old one "DESKTOP-OLD\SQLEXPRESS" rather than your new one "DESKTOP-NEW\SQLEXPRESS". That's why you got the error and in the error, you can see the error hint message below:
ADDITIONAL INFORMATION:
... server attribute must be equal with the true server name.
So, to change your MSSQL server name from your old one "DESKTOP-OLD\SQLEXPRESS" to your new one "DESKTOP-NEW\SQLEXPRESS", first, with two SQL queries below, you need to check your Windows device name and your MSSQL server name respectively:
SELECT HOST_NAME() -- Get Windows device name
SELECT @@SERVERNAME -- Get MSSQL server name
Now, you could check your Windows device name "DESKTOP-NEW" and your MSSQL server name "DESKTOP-OLD\SQLEXPRESS" respectively as shown below:

Next, with the SQL queries below, you can drop your old MSSQL server "DESKTOP-OLD\SQLEXPRESS" and add the new MSSQL server "DESKTOP-NEW\SQLEXPRESS":
EXEC sp_dropserver 'DESKTOP-OLD\SQLEXPRESS' -- Drop server
GO
EXEC sp_addserver 'DESKTOP-NEW\SQLEXPRESS', local -- Add server
GO
Now, you could drop the old MSSQL server "DESKTOP-OLD\SQLEXPRESS" and add the new MSSQL server "DESKTOP-NEW\SQLEXPRESS" as shown below:

But, even if you check your MSSQL server name again, your MSSQL server name is still the old MSSQL server "DESKTOP-OLD\SQLEXPRESS" as shown below. *But no worries, in spite of that, the old MSSQL server "DESKTOP-OLD\SQLEXPRESS" is actually dropped properly and the new MSSQL server "DESKTOP-NEW\SQLEXPRESS" is actually added properly:

If you really want to check that the new MSSQL server "DESKTOP-NEW\SQLEXPRESS" is actually added properly, restart MSSQL with SQL Server Configuration Manager(SSCM). This is How to find SQL Server Configuration Manager in your Windows Machine:

Then, if you check your MSSQL server name again, your MSSQL server name is now the new MSSQL server "DESKTOP-NEW\SQLEXPRESS" as shown below.

Next, if opening "Properties" from the user "john" under "Security/Logins":

Now, you can see that your new MSSQL server "DESKTOP-NEW\SQLEXPRESS" is added to "Securables:" and your old MSSQL server "DESKTOP-OLD\SQLEXPRESS" is removed from "Securables:" as shown below:

Finally, you can grant the permissions for your new MSSQL server "DESKTOP-NEW\SQLEXPRESS" by clicking "OK" without error:
