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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”.