No. They're not claiming that they are a bank. In the English language, bank can also mean a lot of other things. There are loads of sites with bank at the end: letterbank, foodbank... look on Google.
You can see their disclaimer at the footer ... ClickBank is not a chartered bank or trust company, or depository institution. ClickBank is not authorized to accept deposits or trust accounts and is not licensed or regulated by any state or federal banking authority.
How can Google have GO in their name when they are not GO-ing? How can Bing have Bing when they are not BInging? Come on. It is just a name.
One of my favorites banks is the heritage seed bank - nope no money involved, just genetic material saved for use in replanting the Earth after the world disaster of 2012....
They are a click"bank" for the advertisers (They can pay to get clicks/sales). And for publishers as well, they are click"bank" (they can sell their traffic/clicks and get paid).
It varies from country to country. In some countries, it's not permitted to use the word "bank" in a registered business name, trading name or corporate name of any kind without first satisfying the appropriate licensing body/regulator that you're duly licensed as a bank. Other countries have different ideas.