They do, and when you choose to accept bad luck, bad things actually do happen. For instance, if you get bad luck for belongings, your tools will break much more easily. If you get good luck instead, your tools will never break for the remainder of the day (until Isabelle's announcements at 5AM), which is excellent if you plan on spending a lot of hours fishing, catching bugs, or moving flora without having to craft or buy more tools.
For me, I honestly think the friendship luck becomes kind of useless after a while if you have a lot of things in your catalogue and have all your villagers' pics. Good belongings luck is beneficial for me, as well as the health one for building outside (eating food doubles your energy level). The money luck also gives you a higher chance of finding gold nuggets, so there's that.