The fishiness in fish sauce cooks off. Unless you are using an unusually large amount of it, your dish shouldn't taste fishy after it is cooked. It certainly does have a flavor of its own, but is not particularly fishy after cooking.
It is a lot like using anchovies in Italian cooking, where you add one or two anchovies and mash and "dissolve" them in the sauce or dish. Adds umami to the dish. The Romans used a sauce similar to Asian fish sauces called garum. It was highly valued. I often add a bit of fish sauce to non-Asian dishes just as a flavor boost. Works well with stews and any tomato based dish and is way more convenient that digging out some anchovies (I have a bottle sitting with the oils, soy sauces and vinegars close to the stove). As other have noted, you do have to be aware of the overall saltiness of the dish as fish sauce adds quite a bit of salt to the dish.