They just have different criteria for insurance coverage. Sure you paid taxes into it, but you pay your home owner/renter insurance and you pay for your car insurance too. That doesn't mean you get to file a claim for both if you get into a car accident. Different policies are different.
The only problem is you misunderstood the criteria for unemployment insurance coverage. Most know that the coverage doesn't happen unless you lose your job without fault. Most understand that you have to be willing to work (i.e. send out resumes and apply for jobs). Most just don't think about the rule that you have to actually be able to work, but really, why apply for a job you can't do. Or maybe you misunderstood the rules for SSDI. SSDI isn't for if you lost a job due to disabilities, its for if you can't do any jobs due to your disabilities. Either way, applying for both suggests something with the story is off or needs some careful explaining to the people making the decisions. I have heard of people who have successfully applied for both, but I doubt its due to well reasoned arguments, but rather administrative errors, and some have gotten bills from the government when found out.