I understand your point. As mentioned in Murphy's Probabilistic perspective book Kernel's can actually be thought of as a measure of similarity between objects in some abstract space. Probably I should have been more explicit about instead of a user specified feature map we can just use some kernel (ex: RBF). But, I guess it is still clear in the blog that instead of transforming data into a higher dimensional space just the inner products are necessary in the hyper-planes. I will try to modify the text nonetheless, thanks for commenting!