On the question of multiple weapon training, there is some overlap between learning different weapons.
For example, I've trained in the use of a 5'-6' staff. If you break that in half to a pair of 3' Escrima sticks, I can apply a lot of the same basic moves and techniques that worked with the staff. Break one of those sticks in half again and connect the two halves with a short chain, and again the basic staff techniques apply to the nunchaku as well. So if you are skilled with one weapon, it is not too difficult to pick up a similar weapon and make use of it.
Now that having been said, there are also subtle differences between each of these weapons, and they each require specific training to master. For example, I don't have to worry about hitting myself in the back of the head as much with a staff as I would with nunchucks (I've done both), as the potential for overswing is not nearly as great; or the fact that Escrima sticks, being a pair, can be used in configurations that you can't do with a single solid piece of wood (staff).
The point is that while learning the basics of one weapon can help in the use of other similar weapons, each specific weapon still requires specialized training to make full use of its capabilities. So when it comes to realism of game terms, it boils down to a question of what is meant by "proficiency"? If proficiency (as the word implies) is having a basic level of skill, such as being able to swing the weapon around without seriously injuring yourself, then yes, I can reasonably see someone with a martial background (such as a fighter) being proficient in a wide range of weaponry. However, this would not be the same as being a master of such weapons; this has often been marked in previous editions as "specialization", which indicates a higher level of skill, training, and experience with a particular weapon and only that weapon.