What is the intuition behind Ridge Regression and Adapting Gradient Descent algorithms?
Adagrad is completely unrelated to ridge regression.  There is no reason to expect that there should be any similarity in their methods.

Adagrad divides by the L2 norm of the gradient.  Ridge regression modifies the objective function by adding the L2 norm of the parameters.  So even though it looks like the same thing, it's really not; those are two unrelated values.

There's no principle in mathematics that the L2 norm always has to be added or divided.  It's just a number.  It's like seeing one equation where you multiply by 2 and another equation where you add 2 and asking "why does the second one add 2 rather than multiplying by 2?" -- the answer is "why not? there's no reason to expect them to have to use the number 2 in the same way".