Co-operatives

Co-operatives are businesses owned by their workers or customers, who share the profits.
An example of co-operative in the UK is Co-operative bank.

Is the Co-operative bank owned by its customers?

I checked wiki, it says the co-operative bank is a public limited company. I thought there is a difference between a limited company and a co-operative as to ownership and control.
So can a company be both? And is the co-operative bank an example?