Can a Windows batch file determine its own file name?

Yes.

Use the special %0 variable to get the path to the current file.

Write %~n0 to get just the filename without the extension.

Write %~n0%~x0 to get the filename and extension.

Also possible to write %~nx0 to get the filename and extension.

You can get the file name, but you can also get the full path, depending what you place between the '%~' and the '0'. Take your pick from

d -- drive
p -- path
n -- file name
x -- extension
f -- full path

E.g., from inside c:\tmp\foo.bat, %~nx0 gives you "foo.bat", whilst %~dpnx0 gives "c:\tmp\foo.bat". Note the pieces are always assembled in canonical order, so if you get cute and try %~xnpd0, you still get "c:\tmp\foo.bat"