You can't outright spam the court. But you can fill the court with lots and lots of legitimate cases, and lots of people all filing suit against the same company for essentially the same thing is a real thing that does legitimately happen. The US courts have developed some mechanisms - class action, but also a thing called multidistrict litigation (https://en.wikipedia.org/wiki/Multidistrict_litigation) to help with these things.
I've never been personally involved in arbitration, but I wouldn't be at all surprised if the arbitration firms have not developed any real equivalents. Heck, being for-profit organizations, I would think that they have every incentive to not develop anything like that.
In the US, arbitration HAS developed the equivalent of class action cases. However, most companies that require their customers to agree to mandatory arbitration also require them to agree not to utilize class action options.
I've never been personally involved in arbitration, but I wouldn't be at all surprised if the arbitration firms have not developed any real equivalents. Heck, being for-profit organizations, I would think that they have every incentive to not develop anything like that.