Yes. However, the enclosure may not pass that ID through properly, and many USB SATA adapters mangle the numbers or substitute their own ID instead of passing the drive's true ID. Also, some enclosures properly pass SMART info, some don't.
It depends on the enclosure. But, since USB is so bad about connection integrity, there isn't a body of knowledge on what works best, because even when it works the best it can, it's still slow and problematic compared to SATA or SAS.
So, it's possible to get something to work the way you are asking, but you will be dealing with trial and error, mostly error. Is the hassle factor worth it, when the final product is guaranteed to be slower and less reliable than the well established alternative?
How would you cool them? External drives aren't engineered to be run continuously, they will overheat without some sort of forced airflow or other cooling.
All that being said, there is nothing stopping you from forging ahead and trying it. Unraid will definitely let you set it up exactly as you propose, as long as each drive sends a unique serial number that Unraid can use.