Program linked to same folder database

I would recommend you to not use the "attach database through connection string"-feature in the first place.

Reason 1: It's only supported for SQL-Server Express editions. If you want your application to be able to use 'higher' editions of SQL-Server, it won't work anyway.

Reason 2: Microsoft recommends to stop using this feature because they are considering to remove it in a future version.

The alternative is completely trivial: Use SQL Server Management Studio to attach your database, making it a regular, permanent database. If you want to move your project to some other PC you can just either detach it or make a backup and attach it again on the other PC.

Edit after the question was updated:

It's possible to attach a database (an .mdf-file) programmatically. That is, you could let your application perform what I've mentioned in the paragraph above. In combination with my arguments for not using your current approach this could only serve to strengthen your point of proving your knowledge about C#, .NET and SQL-Server.

Please refer to this Codeproject-article:
Programmatically Enumerating, Attaching, and Detaching SQL Server Databases[^]
And, if needed, you could find further information on that with this Google search:
https://www.google.com/search?q=sql+server+attach+database+programmatically[^]

The space in "AttachDbFilename=E:\Programs\C# Programs\Atestat\Atestat\elevi.mdf " is an issue

The problem with connectionstrings is that they have to make sense within the string as well as in your code.

A string is just a string, right?

Well, yes and no. Your program will not know if the syntax within the string is correct. It will just say "yup - that's a string".

The string parser doesn't know when a parameter starts or ends. Sure, everything after the '=' is the parameter value but where does it end?

The answer? Whitespace!
To the parser, it looks like this: "AttachDbFilename='E:\Programs\C#' blahblahblah"

To get around this, the parser accepts single quotes as strings:

"AttachDbFilename='E:\Programs\C# Programs\Atestat\Atestat\elevi.mdf'"

Try it and let me know.

Try this:
Get the executable startup path with Application.StartupPath property, than build dynamically the connection string:

string ConnectionString = "AttachDbFilename = Application.StartupPath + "\\elevi.mdf";