If you are replicating from #1 to #2 then there is no need to query both databases, the data you are trying to access will be in both so just query one of them.
If "NB: the 2 databases can contain different set of data for similar tables." is also true then you only need to query database #2 as all of the data from #1 will have been replicated to #2.
Alternatively, describe these "daily common tasks" you are trying to achieve and you might get better solutions.
However, there are three ways you can access remote databases in MSSQL - all of which are described here
https://www.sqlshack.com/querying-remote-data-sources-in-sql-server/[
^]
Namely (all links are to Microsoft Docs)
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OPENDATASOURCE[
^]
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OPENROWSET[
^] (From your description I think this is what you are after)
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Linked Servers[
^] (possibly in conjunction with
OPENQUERY[
^])