Oh, I could write reams about this! I was the eternal lesson horse rider.
First of all, as you already know, every horse is different and each horse has something to teach us. Many lesson horses, however, are in a class of their own. Different breeds, sizes, shapes, mostly older, often stubborn or cranky, some misused and many improperly trained or de-trained by newbies, poor riders, kids, etc. The list and the possible combinations of variables are endless.
Now, to get to your specific question: whether it is ‘better’ to learn to ride on one horse or multiple horses in terms of learning and muscle development. Well, it depends. (HA! What did you expect?)
As you have said, the other ‘ladies’ you ride with seem to have progressed more in their riding with their own horses. From this comment and your question, I assume you are an adult/older rider (I am!) who is still gaining strength. Correct? Are the other ladies stronger/fitter than you? And, more importantly, what about their horses? Are they stronger, fitter, younger than the horses you are riding? I bet they are, and I bet they also came better trained. So this comparison of your riding versus theirs is completely unrealistic. Like comparing apples and oranges.
I am sure that their younger, well-trained horses are very good at listening to their rider’s aids, have a comfortable trot and always pick up the correct canter lead by simply giving a leg aid. This is obviously true, or these horses would not have been bought. But you? You are having the darnedest time getting your lesson horses to listen to your aids, and you are convinced that this has to do with muscle development. Well, muscle development might have something to do with it (only you know how fit you are), but do you think that a tired, stubborn lesson horse is really going to respond better to more muscle? More importantly, do you think that the other ladies in your class, who are riding the same well-trained horses every day (and ONLY their own horses), would do any better on those lesson horses? I bet not.
When you ride many different horses, you are able to develop a different skill set than single-horse riders. How to read a horse, finesse your aids, find which is his better side/better gait, how he responds to more contact/leg/seat/bend/flexion, etc. Over time, you learn how to adjust your aids, making them clearer or softer, constant if necessary, applying leg, seat, bend, etc. and the horse will know who is boss. And being the boss is BIG with many lesson horses. I am sure you have heard it before, but as herd animals, horses need to know who is their leader. And if the human on their back is not leading, then they take over. And, with a lesson horse, this is GAME OVER.
I work with a trainer, and I do not have my own horse. My riding hundreds of horses over the years have made me a desirable assistant, I guess. I am currently working with a privately owned horse he is training, as well as a children’s lesson horse (weekly tune-ups). Now, the lesson horse gets away with just about anything with the kids, but with me he knows I am all business. To give a clear example, he is older (21) and does not like taking the right canter lead. So, with the kids, he starts with the left and then changes (he is a GP trained Dressage horse). But, I am stronger than the kids, and I’d like to think I’m smarter than the horse. That is when the finesse and clarity of the aids come into play. I do not simply apply a leg-back canter aid. I sit deep, shorten my reins, take him into a corner, more contact, apply my inner leg so he is bent around it, weight to the inside, a bit of flexion… and then he is practically begging to jump out of the corner with the correct lead when I finally give the canter aid. And none of this has to do with a lot of muscle strength. Yes, my core is strong, but I have 3 broken bones in my left lumbar area. It’s just setting him up, with the proper timing, so he has no other option than to do what I ask.
So, in your case, I would work on my mental approach. Read more and watch more videos about the aids and the specific problems you are having (CRK training is good, so is Dressage Mastery by Natasha Althoff, Jane Savoie, and there are many more). Understand the importance of a good seat, contact, etc.
Also, talk to your coach and ask if it is possible to ride the same horse a couple of classes in a row to make some more progress. And, when you are frustrated, tell him/her. Happy clients are paying clients! I am sure they wouldn’t want to lose a client. But, also know that riding the same horse again and again is not the best solution. It is just the easiest one.