Corp responsibility and unemployment insurance carryover

Hello I worked for a company and after 6 year moved on to another company. After 6months I just didn’t feel it was a good and quit. Unbeknownst to me the company that owns us purchased the company I quit my job of 6 year to go to. So I wanted my old job back and they were going through the rehire process and ran into my packet which said I was terminated not quit. So they contacted my old store and said I was not rehirable. So I got the bad news well then I got to thinking if they can carry over that then doesn’t it make them liable for unemployment insurance also. If they act or decide as one then it should all be under the umbrella correct? And if not then would or should that information on my separation only stick with that company?