Insurance Companies - something fraudulent?

Every year I find that I have to change my insurer because they increase my annual premium significantly, and that's without incident in the year, which, usually would give an excuse to them to get more money from their customers.

This year, I went onto one of those comparison websites and noticed that two of the prices quoted, from what appeared to be two different companies, were just trading names but actually belonged to the same parent company so although I rang company 1, they were able to retrieve the quote I had from company 2 and offer me that price, and I had a choice of which of the two company names should appear on my documents.

If one company has access to the database of multiple subsidiaries and can use the information in that way, what choice do we really have in which insurance company we choose? Are insurance companies giving us an illusion that we have choice? Isn't this like an individual pretending to be someone else or disguising their identity to fool others?