How many categories of waves exist?

As written, the question can't be answered definitively.
There can be as many or as few categories as humans wish to define. When we categorise concepts or things, the demarkation of these different types of things is often arbitrary. eg in the electromagnetic spectrum the difference bewteen soft and hard X-rays is defined by individuals - it is not an actual universally agreed distinction; or when you are told that there are six distinct colours of visible light (ie red, orange, yellow, green, blue, and violet) some would dispute that there is indigo between blue and violet. In reality there are an infinite number of colours eg between red and orange there is a reddy/orange colour and between red and the reddy/orange colour there will be a redder reddy/orange etc
In fact electromagnetic radiation (ie a progressing variation in electric and magnetic fields) is like any other wave in that it is simply a repeated oscillating motion that is a means of transferring energy without matter. Think of 'waving' hello or goodbye. Your hand repeatedly moves to the left and to the right around a common (mean) point. Despite the constant movement of your hand, there will be no net displacement of your hand over time. (If you make these movements very small, then is it a microwave that you are sending to the person you are waving goodbye to?)
EM radiation is varying electric and magnetic fields, and mechanical waves (eg water waves, Earthquake waves) have actual matter oscillating about a point as energy travels through the medium.
Then we could also categorise waves by the direction of motion eg transverse waves are where the particle's motion is up and down across the direction of energy movement, or longitudinal where the direction of particle motion is backwards and forwards along the direction of energy motion. Then you can have variations of these eg tortional etc You could also talk of progressive or standing waves. Or gravity waves, light waves, sound waves etc etc
How many categories of waves exist? As many as you want. It will depend on how you want to categorise the waves ie how you wish to define the categories.