Is the field of RL is really stochastic approximation theory in disguise? Is RL just a less rigorous version of stochastic approximation theory?
No, but reinforcement learning (RL) is based on stochastic approximation theory (SAT), and these two fields overlap.
In RL, you typically assume that the underlying problem can be modeled as a Markov decision process (MDP), and the goal is to find a policy (or value function) that solves this MDP. To find this policy, you can use stochastic approximation algorithms, such as Q-learning, but RL isn't just SAT, where, in general, there isn't necessarily a notion of MDP.
SAT is the study of iterative algorithms to find the extrema of functions by sampling from them and under which conditions these iterative algorithms converge. SAT isn't just applied in RL, but it is applied in many other fields, such as deep learning. The paper Scalable estimation strategies based on stochastic approximations: Classical results and new insights (2015) by P. Toulis et al. provides an overview of SAT and the connections with other fields (including RL).
To conclude, RL is based on SAT, but RL isn't just stochastic approximation algorithms, so they are distinct fields. If you want to study e.g. the convergence properties of certain RL algorithms, you may need to study SAT. In fact, for example, the typical proof of convergence for tabular Q-learning assumes the Robbins–Monro conditions. However, you can do a lot of RL without even knowing that RL is based on SAT. Similarly, you can do a lot of SAT without ever caring about RL.