Here "my name" refers to the entire person. Alternate translation: "because of me" or "because he is my disciple" (See: [[rc://en/ta/man/translate/figs-metonymy]])
Jesus means that it is the same as welcoming him. Alternate translation: "When someone ... in my name, it is like he is welcoming me" or "When someone ... in my name, it is as if he were welcoming me"