How exactly does "Visual Studio Version Selector" chooses a Visual Studio version?

There are some version info in the file header. Otherwise it would not be possible for the file explorer to display different document icon on the .sln file with a number:

^{representing .sln files for VS versions 2008/2010/2012/2013/2015/2017/2019}

This number on the icon disappears if you edit the .sln file with notepad or some other text editor that does not preserve the UTF-8 signature (see comment by Paul Groke). In this case also the version selector cannot choose the right version, and you cannot open it from the file explorer. You can specifically open it from within Visual Studio and save the .sln file to fix it. Also see this for more on this issue.

The .sln file must be a UTF-8-BOM file and it must start like this:

[blank line]
Microsoft Visual Studio Solution File, Format Version XX.XX[XXX...]
[description]

The [description] is for example # Visual Studio Express 2012 for Windows Desktop for VS 2012 or # Visual Studio 15 for VS 2017.

The second line is case sensitive but the third line (description) is not. If it is missing, the Selector seems to start the latest VS.

The VisualStudioVersion entry is ignored.

However, for the right file icon to be displayed, the [description] has to be exact and case sensitive.