Welcome to the community, @Andrew!
@MeanMachine41 is totally right! You shouldn't experience any issues with the recognition of the drive on another computer. SATA connections are interoperable regardless of the manufacturer or if it's a mechanical hard drive or a solid-state drive. However, the operating system is indeed tied to the motherboard of the computer, so if you change it, you will need a new genuine OS installation and a clean install of it onto the booting HDD. The new mobo won't be able to boot with the same OS from the old mobo.
Make sure you do a full backup of your data somewhere off-site before the system upgrade. Having a duplicate of your files will keep your peace of mind. Better safe than sorry, right?
Hope this helps. Good luck!
SuperSoph_WD