Insert output into another newly created table
Make use of the OUTPUT clause.

Let's say you need to insert #TEMP1 data into #TAB2 when an INSERT is done on #TEMP1:

CREATE TABLE #TEMP1 
([SERVER] NVARCHAR(200),
[NAME] NVARCHAR(200),
);

CREATE TABLE #TAB2 ([SERVER] NVARCHAR(200),
[NAME] NVARCHAR(200));

INSERT INTO  #TEMP1
OUTPUT INSERTED.[SERVER],INSERTED.NAME INTO #TAB2
SELECT  (SELECT @@SERVERNAME) AS SERVER,NAME FROM MASTER.SYS.DATABASES;


Now see the results in both tables:

SELECT * FROM #TEMP1;
SELECT * FROM #TAB2;


See the product documentation: OUTPUT Clause (Transact-SQL) for more information, including some important restrictions.

If you want to automate this process, you need to write an AFTER INSERT trigger. Please find a very good example exactly matching your requirement in SQL Server: Coding the After Insert Trigger in SQL Server by Vishwanath Dalvi.