Large Portable Hard Drives(0.5-1.5TB): do they work only for one computer?

There may be some confusion between different technologies here so I'll break it down a little bit.

If you are looking to buy a actual hard drive the plugs into the USB port on your computers. Assuming it's your own computers nothing special about them, and they're all windows based... Then you should have no issues swapping the drive between them

If one of the computers is a work computer: the majority of companies disable USB thumb drives these days to avoid loss of critical company data into the wrong hands.

If you are encrypting the data through some software method, you should be able to make a copy of the encryption keys across all your computers. There -could- be issues with the encryption key across multiple machines but that would get software specific and out of scope for the answer.

If some of the computers are windows based and some Mac Based... I don't know to be honest, Macs are supposed to be able to read from thumb drives written by windows machines but I am sorely lacking in experience with Macs. Ether way it's entirely possible there could be issues (I can't rule it out).

If you aren't going with a USB external hard drive, but going to get a detachable SATA drive bay for each of your systems, then yes, there are issues with swapping the drive around as windows will see he drive as "native". With native drives and NTFS windows places a digital key on the drive. You can still swap the drive around but windows might give weird messages that it's doesn't recognize the magic number of the drive.

This is a lengthy explanation, but I put the one I consider probably the most relevant at the top. Hope it helps!