I'm a psychiatrist. I'm not going to discuss the relative benefits of anesthesiology as a career versus psychiatry, but I will share the career track of one of my former residents just as a point about what can be done in terms of carving out your own niche.
The two are very different specialities and the training experience will be very different, but one of my former residents did take a career track that broached a gap between anesthesia and psychiatry in an interesting way. As someone noted already, one point of overlap is pain management. Both psychiatrists and anethesiologists do sometimes specialize in pain management. The skillsets that the one has versus the other with respect to pain management are going to differ to some degree. Both can, of course, use pharmacotherapy to address pain. Psychiatrists have in their toolkits, however, evidence-based psychotherapeutic approaches to pain management, e.g., CBT or mindfulness-based interventions. There is a lot of comorbidity between pain and psychiatric and social stressors, so your average psychiatrist might be better equipped to assist with these in a more holistic/person-centered sort of way. However, in general, the anesthesiologist is going to be better equipped to take procedural approaches to pain management. I train psychiatry residents in my role as faculty at a medical school. One of our former residents completed an interventional pain management fellowship after completion of his fellowship. Since completing his fellowship, he performs minor spinal procedures as part of his practice. Needless to say, this is highly unique, but I guess it can be done!
One of the very cool things about medical fields is that there is often a way to carve out your own unique niche or specialty if you want to and are creative. For example, I know many non-psychiatrists who have worked their way into psychiatric or psychiatry-adjacent fields, from Dr. Robert Laitman, who is a nephrologist by training but became a clozapine expert, to many non-psychiatrists who work in addiction medicine. I don't think it's the easiest or most straightforward route to take and it would take some commitment and energy to do something like that with psychiatry, but it can be done. The borders between fields are to some degree artificial. Really, there is one human body and one person and the organ systems all interact.
You still have to choose a residency, though! But I thought I'd share, in case it's helpful.