The amount of times you should climb per week really depends on you and how conditioned your body is to climbing. If you have been climbing for a few years, and you are generally healthy, you may be fine climbing 4 times per week or even more. If you are a new climber, climbing 3 times may be your limit before you start experiencing slight or major injuries, and you may even be able to climb only twice a week.
In the end, the amount of times you can climb per week really depends on you. If you come from a background in sports, and you are a new climber, then maybe you can even push it and go for 3-4 sessions per week, but this is considered risking it because climbing generally uses very specific muscles that most people, even those coming from other sports background, are not used to, and if you overuse these muscles and don't give them the adequate amount of rest, you could just be risking injuries.
The best way to find out is to just experiment. Try going twice a week while having a rest day or two in between, if you are ok with this, try adding another day in, and just keep experimenting with the amount of climbing days and the amount of rest days in between.
There is no definitive amount of times someone can climb per week. It just depends on each person, as each person has their own fitness level and athletic history.
When I started climbing, I climbed around 4 times per week, each session was 4-5 hours long. Needless to say, I injured myself. I toned it down a lot to just 2-3 times per week at most, and also took some time off here and there to give my injuries and muscles more time to recover.
After this, I create a better schedule of climbing 2-3 times per week, for around 2-3 hours each time. I came from a background of weight training for quite a few years. This did help with my climbing as I was able to pretty quickly progress to v6, but, I did overdo it in the beginning. With my new schedule, I was injury free and haven't been injured since.
So even if you have background in something, or you are generally fit, you can still very easily overdo it. Climbing is different from most sports, it uses your muscles differently, so it is just very easy to overdo it.
The best way to know how often you should climb per week is to just listen to your body, pay attention to it, and don't ignore the signs. If you do, the next time you get another sign may be too late.
