I don't think requirements change nearly as often as people think they do. Open up your repository and look at the oldest files... probably still running in production, and never needed any changes. (Additionally, a careful design will still be the right one even in the face of changing requirements.)
Making changes backwards compatible is fairly simple. If semantics change, give it a new name. If mandatory fields become optional, just ignore them. (This is why you use something like protocol buffers and not JSON. It's designed for the client and server operating off of different versions of the data structure.)
Having two ways to do something is always going to be a maintenance burden, and make your service harder to understand. It is not related to microservices or monoliths. The same problems exist in both cases, and the solution is always the same; decide whether maintaining two similar things is easier than refactoring your clients. In the case of internal services where you control all the clients, refactoring is easy. You just do it, then it's done. In the case of external services, refactoring is impossible. So you maintain all the versions forever, or risk your users getting upset.
Making changes backwards compatible is fairly simple. If semantics change, give it a new name. If mandatory fields become optional, just ignore them. (This is why you use something like protocol buffers and not JSON. It's designed for the client and server operating off of different versions of the data structure.)
Having two ways to do something is always going to be a maintenance burden, and make your service harder to understand. It is not related to microservices or monoliths. The same problems exist in both cases, and the solution is always the same; decide whether maintaining two similar things is easier than refactoring your clients. In the case of internal services where you control all the clients, refactoring is easy. You just do it, then it's done. In the case of external services, refactoring is impossible. So you maintain all the versions forever, or risk your users getting upset.