Nor is a union the economic or legal definition of a cartel. A union is closer to creating a company that acts as a negotiating and protective apparatus for its employees as they do contractual work for other companies.
That isn’t a cartel and there can be multiple, competing unions working for the same type of workers in the same industry.
an association of manufacturers or suppliers with the purpose of maintaining prices at a high level
That's the definition of cartel. Unions exist to maximize the amount they extract from buyers of labor. This is rent seeking plain and simple.
What's more, they don't compete with each other, which is what corporations must do. Why does the UAW get to enjoy a monopoly on the sale of autoworker labor? Should Ford and GM and Chrysler be able to unionize together to keep wage costs lower?
You absolutely can. Managers will push back with "rules" that only apply if management doesn't want to pay you more. Or they will go to HR to get an exception if they think you are worth that exception (that is, if they aren't worried about not being able to match an offer for an employee that they really care about). You can absolutely negotiate.
In the past, I've been quite open when I thought that I needed more money to my manager, and have even given specific ways of making me "not distracted by money concerns". Sometimes they can meet those goals, sometimes they can't.
Personally, the offer as you've given it is probably more adversarial than I'd prefer. Something like "I feel like I'm worth more to the company than X, I feel like I'm worth Y, and here is a list of reasons, here is my career goals, etc etc". Then if they don't match it, you can accept that other offer. But YMMV.
