They absolutely should not. Spurious copyright claims are not the answer. Firewatch has had an express license on their site saying that they allow Let's Plays. They can't go back on that, which means that they're abusing the DMCA strikes of YouTube, putting them on par with Alex Mauer et al. DMCA is not a system that should be abused. In the case of Firewatch, who explicitly has written that Let's Plays are allowed, have no legal foundation to file a DMCA strike. Games that haven't, would have to take the legal stand-point that Let's Plays are in fact copyright infringement, and once and for all take the legal battle if Let's Plays can be allowed under Fair Use.
I don't think anyone wants that to go to court, and if that's not the intent of whoever files a DMCA strike on a Let's Play, then they are absolutely filing spurious copyright claims, which is illegal. If it isn't illegal, they're abusing the law to differentiate. That's not what DMCA is for. This should not be condoned.