As stuartd suggested correctly, Server Explorer in Visual Studio is confused. Sending an answer in case it also helps someone else.
I realized that every time I added the Entity Data Model Model Entity Data project for the Entity Framework to the model, as shown below, it added the connection to the Server Explorer window.

The connection corresponded to the database that I selected in the Add ADO.NET Entity Data Model Wizard. I see that there is a red cross against some of them, since I deleted the databases from the instance of SQL Server. Whenever you download a solution, Visual Studio will always try to update the connection of all the databases found in the server browser. In fact, in the case of databases that have a red mark against them (since they are deleted), I am sure that VS should also use retry logic when trying to establish a connection to the database. All this takes and spends time loading the solution.

I deleted those connections marked with a red cross to get rid of the problem. It will be great if someone knows a way to mark connections as inactive, i.e. Connections live in Server Explorer, but VS should not try to connect them every time until I ask him to update or maybe it can update when I span its node.
I also noticed that Visual Studio somehow supports these server explorer connections based on each solution file, maybe in * .suo files. I had to remove these legacy connections after opening every solution that encountered this error related to the problem of connecting to the database when loading into Visual Studio. All the solutions to this problem somehow had at least one project related to the entity infrastructure :)
RBT
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