Most of cantons, if not all of them, would have a cantonal bank on their own, and their logotypes are sort of similar. It makes me wonder if there is some kind of cooperation, a mutual recognition or any kind of link between them, or they are totally independent banks?
For example, would it be possible to change one cantonal bank for another when moving cantons with less bureaucracy?

They are independent banks in most ways, but they have a union (Verband Schweizerischer Kantonalbanken, a.k.a. VSKB) with lots of formal and informal guiding and administrative powers over its members. 24 cantonal banks, to be precise. From Wikipedia: The fact, that you already have an account with one cantonal bank may or may not help you to open an account with another. Depends on if you were a good boy or a naughty one. But with a simple current account there shall be no difference at all.

I'm not particularly sure, what are you referring to as "less bureaucracy". I had never come across much of it with local banks in the first place. Comparing to BofA, RBS and HSBC (outside of CH), who all asked for a lot of forms to be filled and data to be provided, Swiss banks were all but paper pushers with me so far. Only reqestes my ID, a couple of simple forms and, of course, my money. Also asked me in several different ways the same AML question: "are you a US tax resident, a terrorist, a a PEP or in any other way a potentially problematic client?". No more questions asked. Literally an account opening here takes less then half an hour, if you pass on chit-chatting and coffee. But this experience may vary, so would love to here other opinions and stories.

Thank you for a comprehensive answer!
Speaking of the bureaucracy, yes, indeed opening an account is easy, however I am concerned if it'd be possible to simply "relabel" an existing account in one cantonal bank into another, without re-registering, transferring savings etc. Probably not

In the same way there is cooperation between the Swiss National Bank and the Bank of England, there is cooperation between the cantonal banks. Same principle.

Why do you need to change bank all together? Does your cantonal bank require residency in the Kanton? If not just leave the money there?

I imagine there could be a fee when using a different ATM, or when willing to come to an office in person to handle something.

BUT it is not about this particular issue, it is not really an issue at all, I was curious to learn in general about their relations and if there was anything in it for their clients.

There is no fee when using the ATM of one Kantonalbank when you are a customer of another Kantonalbank. In this respect all the Kantonalbanks act as if they are one united bank.

But I don't know if this also applies to personalized services. I guess not. So if for any reason you want to discuss something with your bank manager in person, you might have to travel back to the home canton of your bank.

This is interesting. Thank you!