Depending on the system you are running/playing the number of combat skills probably greatly fluctuates. FATE, for example, has only one or two depending on how you are using it while Legend of Five Rings has about ten all things considered. These skills govern not only who can fight (you need a skill to fight well, after all) but also how that person fights. It is an important thing to consider when making a game too because where you have variety you are making player choice matter, and that should hold consequences both good and bad for the player character in question.
How Many Skills Do I Need?
This is likely the most burning question and the answer is that it depens. FATE gets away with two combat skills (Shooting and Fighting) because the emphasis in FATE isn't to be about a world where combat is all that matters. Combat is just one of the things you can do in FATE and so the system's default skill tree doesn't go nuts trying to chop it down into pieces. On the other hand, L5R has a different skill for each type of weapon (swords, knives, spears/polearms, bows, heavy weapons, unarmed, chain weapons, peasant weapons, ninja weapons) as well as a defense skill for other folks to take.
The reason for this is that in L5R you are playing a member of the samurai caste, combat is common and often, and what weapons a samurai can and can't use is a matter of choice, taste, and in many cases clan or school. Techniques characters get may make them more powerfulwith one specific weapon or category of weapons, and general Rokugani etiquette controls other things (i.e. Ninja Weapons are illegal and can get you killed for knowing how to use, outside of war time heavy weapons should not be used on Samurai.)
This boils down to the question to ask yourself when you need to know how many combat skills your game should have: how important is combat to the game, and how important is the player's choice of how they fight to the game?
Choice Should Matter
Let's say you've decided that you want multiple combat skills because you want the player's to choose how they engage. That is awesome and good, but you have to make that choice matter.
For example, in FATE it matters because you have Shooting or Fighting. Shooting lets you attack from range with guns, Fighting lets you attack close up. The choice then matters because with Fighting your attack skill is also your primary defense skill (you roll it for offense and defense provided you're in melee) but with Shooting you need another skill for defense in exchange for the ability to attack from range.
Another common distinction is the difference between "hand to hand" and "weapons." The idea here being that a weapon generally does more damage but can be taken away while a person's fists can't really be disarmed.
These are both fairly small distinctions, but they do have their place and it would be up to the GM to make the difference between the two.
As a negative example, for a mod I made a while back I broke melee combat into three skills: hand to hand, bladed, and non-bladed. This really should have been hand to hand vs. weapons for the above reasons, but we went further than that for reasons I no longer remember. The end result of this is that players wanting melee abilities are now weakened. How? Because they are making a choice between three items (melee, bladed, non-bladed) but there is no significance to that choice. The choice with significance is: can your melee option be taken away but has higher damage output?
In the end...
How many combat skills your game has is a choice, but it is a choice that speaks to the nature of the game. More narrative and streamlined games need less while more crunchy or combat focused games will need more. The ultimate point here though is that when you present multiple skills for the same thing you are making the player make a choice, and that choice should be meaningful in some way. If there is a different skill for fighting with a pistol than fighting with a rifle then the fact that my character fights with Pistols should be meaningful in some way to the game as well. That meaning doesn't have to be huge, but it should be distinctive.
Ultimately it boils down to this: Choices should have pros and cons tothem, that is how they find meaning.