I’m struggling to figure out what is it that you actually want. You can consider induction on arbitrary well-founded relations instead of ordinals (and only on well-founded relations, as induction actually implies that the relation is well-founded). If the relation is reasonably encoded, its induction scheme should have the same proof-theoretic strength as induction on the ordinal which is the rank of the relation. Thus, the only thing you can achieve is to have ordinals represented nonuniquely by a fancy, more complicated structure. As a matter of fact, this is what you do anyway, since e.g. in the usual representation of ordinals below $\varepsilon_0$ in arithmetic using Cantor normal form, ordinals are identified with certain trees. So the answer to your question appears to be “yes, just call them trees instead of ordinals”.

I’m struggling to figure out what is it that you actually want. You can consider induction on an arbitrary well-founded relation instead of an ordinal (and only on well-founded relations, as induction actually implies that the relation is well-founded). (This covers all the various special cases like transfinite induction, structural induction, $\in$-induction, and whatnot.) If the relation is reasonably encoded, its induction scheme should have the same proof-theoretic strength as induction on the ordinal which is the rank of the relation. Thus, the only thing you can achieve is to have ordinals represented nonuniquely by elements of a fancy, more complicated structure. As a matter of fact, this is what you do anyway, since e.g. in the usual representation of ordinals below $\varepsilon_0$ in arithmetic using Cantor normal form, ordinals are identified with certain trees. So the answer to your question appears to be “yes, just call them trees instead of ordinals”.