There is a relation between scaling and renormalization, but not between scaling invariance and renormalization.
Scaling is possible for every polynomial Lagrangian density. In scaling, one transforms space-time, fields and coupling constants by suitable powers of a dilation factor in a way that preserves the action. Scaling is the reason for the existence of the renormalization (semi)group. It implies that the parameter defining the mass scale in a renormalization prescription is redundant.
On the other hand, scaling invariance means that the coupling constants are independent of the dilation factor, which is a rare situation. For example, QED is renormalizable and has a scaling operation affecting the electron mass, hence is not scale invariant.
Self-similarity is a property of fixed points (critical points) of a renormalization group mapping. However, in QFT this self-similarity is not a property of a physical theory but of the regularizations at different energy scales. (This energy has nothing to do with the mass scale in the renormalization prescription, though some formulas at 1 loop look very similar.)
This post imported from StackExchange Physics at 2014-03-12 15:42 (UCT), posted by SE-user Arnold Neumaier