How many running shoes you keep in your closet is up to personal preference, but also depends on how much running you do each week and what kind of running you are doing. For runners who run a LOT (like elite runners, professionals, or those people logging a ton of miles as they train for high mileage races like marathons, ultra marathons, and triathlons), multiple shoes are definitely a wise investment. Swapping out which shoes you run in helps preserve your shoes for longer, and switching up the "feel" of the shoes you run in can actually help make your feet stronger as they have to adjust to different insoles, different layers of cushioning, and the amount of support.
Having multiple options of shoes is also a good choice if you plan to run different types of runs. For instance, the shoes you wear on a long trail run are not what you want to wear the next day during your 400 meter repeats on the track. If you are changing up the style, pace, and distance of your workouts throughout the week, then you need several shoe options to accommodate different terrains and speeds.
A lot of runners prefer to have at least two different running shoes that they run in. One can be saved only for long runs outside, and the other is usually used for shorter tempo or speed drills on a track or paved surface, as well as for indoor runs on the treadmill. The shoes worn on the track and indoors are more of a minimalist shoe, meaning they are lighter and help the runner's foot turnover to be faster. The long run shoes have a thicker sole and a lot more cushion, stability, and support to help keep their feet and ankles from getting injured on uneven terrain, and so they can last longer even when exposed to mud, dirt, water, and the elements. So in short, no, having three pairs of shoes is not a bad idea and can actually help benefit your running immensely.