What type of adhesive should I use to attach metal to vertical tile?
If I had to install something like that without mechanical fasteners I'd try a product called Lexel. It's silicone based so it's good in wet conditions and sticks to pretty much anything, but it gets harder than traditional silicone so it forms a bond that's scary strong. Having said that, relying solely on an adhesive in an application like this is not recommendable. Ceramic tile is really good at not letting stuff stick to it,  and a falling knife rack is the stuff of my nightmares. A couple hollow wall anchors will buy you a lot of piece of mind.