Before you started working out, your muscles weren't dense at all. Eventually, through your bodyweight workouts, your muscles become denser, harder, and fill out. This is called newbie gains, and it is possible to achieve with bodyweight exercises. Some people have been known to put on 20-30 lbs during their newbie gain phase.
However, after the first year, your muscles will be filled out and dense. You can't make newbie gains like before, because your muscle tissue is as dense as it is gonna get, and your body has to build new tissue. Those types of gains are SLOW.
Now we're gonna get into resistance training. The human body is an amazing organism, it adapts to anything you throw its way. Right now, your body still hasn't gotten used to the push-up routine, and your muscles are still getting denser and denser. Eventually, your body adapts to your routine, and push-ups become easier and easier. Once that happens, you stop making gains from your push ups. You can up the reps, but you will make minimal size strength/size gains from this. All you will get is endurance (which is useful in martial arts). The key is adding resistance, either by wearing a backpack filled with weights while you do the push ups, or actually getting on a bench and doing some presses. It is only then that you will be able to stress your muscles and force them into growing again.
Next, the problem with bodyweight exercises is that you do not work out all your muscle groups. Your lower back, side and rear deltoid, trapezeoids, serratus wont get much workout. Your quads will get MINIMAL gains from bodyweight squats, and your calves will remain the way they are now. Your arms will not be that great either, as there is only so much you can make them grow from chinups / dips / push ups. Your back will widen a bit from the pullups, but it will never have that thick look.
Why is this a big deal? First of all, you will have a disproportional body.
Second of all, you are missing out on the indirect effect. Arthur Jones, the man who designed the first Nautilus machines discovered this in the 1970s. When a muscle grows in response to an exercise, it will stimulate growth in other muscles. Big muscles will stimulate more growth then smaller ones (hence why squats are so important, as they stimulate a huge muscle group which in turn produces gains in other areas of the body). The more muscles you leave out of your workout routine, the less you will grow. You will notice that people who tend to leave squats out of their workout routines usually never have the size and girth of a guy that has a complete workout routine. Not just that, but they have chicken legs (but thats another story)
With a bodyweight only program, you are working out perhaps 60% of your body. By not stimulating all of your muscles directly, you are not only making minor gains that will make your body look incomplete, but you are also making SLOWER gains on the muscle groups that you are currently working out!
I'm not saying your trainer is teaching you the wrong stuff, but you should educate yourself when it comes to weight training. You should be open to ALL training ideas, not just the ones your coach tells you.