[Comment: what follows is not really an answer, but rather a focusing of the question.]

In general, there is not such a nice description even of the number field $\mathbb{Q}(a,b)$ -- typically it will be some non-normal number field whose normal closure has Galois group $\operatorname{GL}_2(\mathbb{Z}/n\mathbb{Z})$, where $n$ is the order of the torsion point.

In order to maintain the analogy you mention above, you would do well to consider the special case of an elliptic curve with **complex multiplication**, say by the maximal order of an imaginary quadratic field $K = \mathbb{Q}(\sqrt{-N})$, necessarily of class number one since you want the elliptic curve to be defined over $\mathbb{Q}$. In this case, the
field $K(P)$ will be -- up to a multiquadratic extension -- the anticyclotomic part of the $n$-ray class field of $K$.

And now it is a great question exactly what the rings of integers of these very nice number fields are. One might even venture to hope that they will be integrally generated by the x and y coordinates of these torsion points on CM elliptic curves (certainly there are well-known integrality properties for torsion points, although I'm afraid I'm blanking on an exact statement at the moment; I fear there may be some problems at 2...).

I'm looking forward to a real answer to this one!