OK, let's be clear here.
Changing out a HDD for another HDD in the same computer does not make the operating system think it's in "a new machine."
What you have been told is accurate, as far as it goes. Were you trying to use a different machine than the one that Windows (I presume) and all your application programs are now installed on you would need new license keys for Windows itself (or to have the one that was related to the "new" machine if it's actually "new to you" if you're reinstalling Windows and the original license was retail, not OEM) and for all the various programs.
If you are simply upgrading (or even downgrading) your HDD, so long as the target HDD in the clone process is large enough to handle all of the content from the source HDD you just need to clone the original source to the new target. That new target is "plug n' play" so long as the machine it's being plugged in to is the one from which the source HDD was obtained.
You do not clone an old drive to a partition on a new drive. The cloning process duplicates, exactly, what is on the old/source drive on the new/target drive. After that process is complete most cloning software will also allow you to repartition the target drive to take advantage of the additional space you have if the target drive is larger than the source drive. If you use something that doesn't allow this as part of the cloning process, then you can use the partition manager of your choosing to do the repartitioning after the newly cloned drive has been installed.
If you have additional questions with regard to drive cloning I suggest you review some of the recent content in the Backup, Imaging, and Disk Management Software forum and, if your question is still unanswered then post a new topic there. We're reaching well beyond "Tips n' Tricks" now and delving deep into disk management/imaging.
Edited by britechguy, 03 March 2018 - 11:06 AM.
Addendum about Backup forum
Brian AKA Bri the Tech Guy (my website address is in my profile) Windows 10 Home, 64-bit, Version 1709, Build 16299
Here is a test to find out whether your mission in life is complete. If you’re alive, it isn’t.
~ Lauren Bacall