It may be possible, but not likely practical. You'll get a user license error unless you figure out a way to install two copies with different license keys on the same machine.
But simpler/cheaper alternatives for this are possible:
1) Close all but one window in file 1 when opening file 2. This keeps the file open while reducing the "clutter". I generally try to keep no more than two or three windows open and visible at one time inside a windows filemaker application specifically to reduce clutter.
2) Instead of closing the windows minimize them, or reposition them to a location with negative coordinates...
3) Go ahead and close the file. If you define matching account names and passwords in each file, a script from File 1 can open File 2 and FileMaker will automatically use the credentials that opened file 1 to open file 2 without asking the user to log in.
Thanks for your quick response and helpful suggestions! I'll try these and see what works well here.