This has the appearance of anonymous when it is not.
First, the moment that a value, being it a nonce, a random value or whatever will be common on both side, there is no anonymity anymore.
Then, there is timing attack, where visiting a website, you then need to go to your bank. And the bank will sign at a specific time for a specific timeframe.
Then, the need to manually "copy" the signature. I guess you don't see what size, difficult characters this has to have, but totally impracticable. So in the end people would use that to pretend that it is possible with anonymity skipping that step.
Then, if the bank doesn't know the website where you went, gambling.com will know what is your bank.
And in a lot of countries, age limit might be different for some activities, like gambling allowed at 21, porn at 18, or even rules would not be the same by countries. So again, you will be leaking the country and potentially you will have to leak to the bank the activity that is intended.
And what do you do when banking app force you to have a certified Apple or Google spying approved phone? Bank already have too much power and responsibility and easily abuse of it, so it would be better to not give them even more.
So again, please refrain of inventing stupid solution like that, that can give the wrong impression to the bad persons that it is possible, justifying laws and co, when it is not possible without costing hardly on our privacy!