I have tried using the license key that I have on my desktop on my laptop, but it only gave me a trial version.
Had you previously installed the trial version on your laptop? Don't see how else that would happen. You should uninstall the trial version before installing the licensed version.
Read your license agreement on what you are permitted to do with FileMaker. As I understand it, you are permitted two installs as long as you only use one at a time.
From a techincal perspective, there's no barrier to installing the same software on two different machines of the same operating system. But if you host a database from one computer, you will be unable to connect to it as a guest from the second computer unless it has a different license key.