The main governing law regarding online services and copyrighted content is the "Online Copyright Infringement Liability Limitation Act" (OCILLA), or DMCA Title II, commonly known as "DMCA takedowns". It provides a legal requirement for online services to take down infringing content and ban users who frequently accumulate takedown notices. The process is actually suprisingly balanced and fair for being part of US Copyright law - there's a counter-notification process which requires any further legal action to occur as actual legal action, it's very easy to properly comply with OCILLA and get your safe harbor protection (unless your name is Kim Dotcom), and more importantly, there is no legal requirement to proactively monitor your services. That last one is important, because it's what we fought SOPA/PIPA over - user-created content can't exist if the law doesn't recognize that users are creating the content, not the services; moreover, without such recognition the Internet would quicky become less like the Internet and more like cable TV.
For most Internet services, the DMCA is enough. If you're, however, YouTube, you have your own parallel takedown system called "Content ID". First things first: Content ID is not a takedown system: It neither requires content owners identify infringing content nor does it restrict it's actions to only takedowns. Why YouTube built the Content ID system is a mystery - they aren't legally required to. My current intuition is that it was to attract high-value premium video content (read: cable and network television) to YouTube. YouTube had a reputation of being used primarily for unauthorized content and content owners didn't want to legitimize the service in it's current form. As such, Content ID was put into service to "clean up" YouTube no matter what the cost.
Content ID judges infringement not based on legal claims of the content owner but by content similarity as judged by a computer vision algorithm. The DMCA lets content owners claim infringement while Content ID lets content owners claim ownership and the system sorts out the hairy bits (in theory). Additionally, what's done to claimed videos depends on a whole host of factors. DMCA only allows worldwide takedowns, but Content ID lets content owners either block the video, get access to it's Insight statistics, or monetize the video by placing ads on it. The claim and it's effects can be limited to certain regions - or only be limited to blocking audio, in the case of music rights claims. For example, GEMA is a music licensing agency responsible for blocking or muting many thousands of YouTube videos (even ones that are properly licensed elsewhere) for German country users.
Now with the advent of Nintendo mass-claiming revenue from Let's Plays and actual game reviews we've had the revelation that several other prominent game developers, including Mojang, were offered the ability to place a Content ID monetization claim and get all the revenue from video of their respective games. Naturally, most game developers flat-out refused, because Let's Plays and other creative video reuse have the function of massively increasing sales.
Let's just get this straight: what YouTube is offering is completely wrong. It's one thing to say, "Hey, a bunch of people are reposting your videos or using your audio without permission, would you like us to slap some ads on that?". It's another thing entirely to say "Hey, a bunch of people are reusing your copyrighted in a work which would most likely be fair use if it came down to a court decision, would you like us to slap some ads on that?". I'm personally of the opinion that Let's Plays, like reviews and criticism, should be considered the property of the person who created the video and not the person who created the game. YouTube should not be in the business of monetizing other people's work without permission.
The difference between the two scenarios is still murky in the eyes of the law: ostensibly, copyright law has a provision for "derivative works" which includes any possible reuse of a work, not limited to but including adaptations and translations. Copyright caselaw, on the other hand, developed a notion of "fair use" which includes certain creative reuses of a work which would otherwise be classified as derivative works, not limited to but including parody, review, criticism, and educational uses. Under most reasonable interpretations of fair use caselaw, at least some kinds of Let's Plays would be considered fair use and not subject to the rightsholder's monopoly on derivative works.
Think about it this way: If you and your friends were to play baseball, record it, and post it to YouTube, should Major League Baseball be able to claim ownership over it and take it down? No - the only thing they created is the MLB ruleset. Let's slippery-slope this a bit: what if we replace Major League Baseball with a TCG like Magic: The Gathering or Yu-Gi-Oh!? Now there are visual representations of the cards that are copyrighted, yes, but the point of the video isn't the cards, it's the people playing the game. If we go further and replace the TCG with a full-on videogame then you could argue that there's additional visual representations of gameplay that are owned by someone else. But the same logic still applies - it's a video of gameplay, not the game.
My own moralizing and legal theories aside, however, it's more interesting to wonder: why is YouTube the one initiating contact with game developers? Given how much Google hates the idea of having to proactively monitor it's search for pirated content, you would think they'd be against any proactive monitoring beyond what they had to in order to get content deals. In fact, they even foregoed content deals with the music labels for a good year because Google didn't want to start filtering search results. It may be the case that Google lets YouTube do it's own thing.
But my personal conspiracy theory is that YouTube is intentionally pushing for Content ID usage in order to drive advertising revenue. Remember first that YouTube content isn't monetized until you're approved for monetization or a Content ID owner makes a claim on your video. The result is that a large majority of YouTube content is unmonetized. YouTube itself, obviously, would like to see that change. Things like wider access to channel-wide monetization and single-video monetization offers help, but that only goes so far. Not everything on YouTube is 100% clearable for monetization and a fair amount of it's videos are completely unlicensed.
Note the way YouTube phrased the system to game developers. It's not "keep unlicensed use of your work off YouTube", but "monetize unlicensed use of your work on YouTube". They are approaching the situation as a way to sell more advertising. So here's my hypothesis: YouTube aggressively pushes Content ID specifically as a way to be able to monetized videos that shouldn't have been uploaded in the first place. They definitely do want to profit off of unlicensed video usage, and they do it by getting people to put their work in the Content ID system and monetize it.
I honestly think that kind of system is wrong, for both artists that are trying to sell video or audio content, and people making new transformative works finding their work being tarnished with ads they don't want. Remember that this isn't the first time this has happened (the Nintendo thing). EA started claiming Battlefield 3 videos last year, and someone who made a mashup parody making fun of the unfortunate implications of the Twilight series got the same treatment.
Actually that last case is a more compelling example of why YouTube itself is a horrible video sharing platform. The short of it is that someone made a gender critique of vampires and didn't monetize it. YouTube monetized it for them, through Content ID, and slapped fashion ads all over the video pointing out misogyny. The problem here is that not everybody wants ads, especially over things which could be used in a Women's Studies class! As far as I know, YouTube is the only video host that has a Content ID-style system. Everyone else relies on the DMCA takedown process. Sure, yes, there are automated DMCA takedown bots too, but those require the copyright owner to actively decide to use one. The bigger thing, though, is that Content ID makes copyright-related harassment easier. In DMCA, once you've been counter-claimed, you either engage in expensive litigation or you drop the matter. But with Content ID it's much harder to get to that point, and once you do so you've already taken away their ability to upload longer than 15 minutes and made them watch an absolutely terrible "copyright school" video.