Sql-server – When should you split a database into 2-3 databases for performance reasons

Ask the developers/architects who designed the solution. Only they can really tell. I can think of several reasons:

1. It could be that the databased was added at different times. First the main database, then they decided they needed a database for the events and last a place to store old data. Depending on environment it might be easier to create a new database rather than adding a lot of new tables to an existing one. I've worked on systems where they at times took a backup of the production data, put it online as a read-only archive and then removed all old data from the current db - thus having several old databases available at the same time.

2. Future use: maybe they thought that the system eventually would have to be split up on different servers (or perhaps just faster discs for that high traffic database).

3. Plans for different back up times etc. Maybe the archive is never down for backup but available for statistics 24/7, production is perhaps backed up daily or even more often and some other plan for the event logs?

It could actually be all, some or none of the above...

Your problem begins and ends with statistics and estimates. I have reproduced your situation on my servers and found some interesting hints, and a workaround solution.

First things first, let's take a look at your execution plan:
When a view is used we can see that a filter is applied after the Remote Query is executed, while without the view there was no filter applied at all. The truth is that the filter was applied inside the Remote Query, at the remote server, before retrieving the data over the network.
Well, obviously applying the filter at the remote server and thus retrieving less data is a better option, and obviously that only happens when not using a view.

So... what is so intersting...?

Surprisingly, when I changed the filter from cognome = 'test' to cognome = N'test' (unicode representation of the string) the view used the same execution plan as the first query did.
I guess the reason is that somehow when using the view SQL Server estimated that there will be a small number of rows returning from the (remote) query, and that a local filtering will be cheaper, but when SQL Server had to implicit convert NVARCHAR to VARCHAR, statistics could no longer be used and the decision to filter locally was not taken.
I have looked for the statistics locally, but the view had no statistics, so my guess is that the view uses the remote statistics in a way that ad-hoc query does not, and than takes the wrong decision.

OK, so what solves the problem?

I stated earlier that there is a workaround (at least until someone comes up with a better solution), and no, I don't mean using unicode for your strings.
I wanted to give an answer first, I still have to find why, but when using an Inline Function SQL Server behaves exactly the same as with the query (without view), so replacing the view with the function will give the same result, in a simple query, and with good peformance (at least in my environment).

My code suggestion for you is:

CREATE FUNCTION fn_anagrafiche2()
RETURNS table
AS
RETURN 
(
    SELECT * 
    FROM dolph2.agendasdn.dbo.vistaanagraficagrp
    UNION
    SELECT * 
    FROM dolph2.acampanet.dbo.vistaanagraficagrp
    UNION
    SELECT * 
    FROM municipio2.dbnet.dbo.vistaanagraficagrp
)
GO

The query will then be:

SELECT * 
FROM fn_anagrafiche2()
WHERE cognome = 'prova'

This works on my servers, but of course test it first.
Note: I do not recommend using SELECT * at all, as it is prone to future errors, I simply used it because it was in your question and there was no need for me to change that when I can add this remark instead :)