I have not seen a lot of anime, but some of the ones I have seen have pocket watches in them. Noir, Phantom, Fullmetal Alchemist, and Steins;Gate all have pocket watches of some sort in them and are usually an important piece at some point in the anime. How come many anime include pocket watches in them? Thanks.
I'm not an expert on Japanese culture but if I had to guess, there are some symbolism involved.
1) A pocket watch became popular after World War I and it was one of the expensive items being imported into Asian countries. It was valuable and was something that can be inherited.
2) A watch symbolizes time; past, present, and future. It represents the character's past (usually dark and sad in anime), present (struggles) and future (final goal, whether it's a bright one or inevitable death.)
That's what I think.
In addition to what monty_bean said, a watch does show the symbolism of time, but it's far more significant with a pocket watch, which can have more meaning, such as the engraving on the watch in FMA.
A wristwatch just does not carry the same emotional gravity that a pocket watch does; it would register more as just a fashion accessory or something to tell the time with. The significance of the watch itself wouldn't be nearly as present.