If we are only talking about this new console generation, I think Microsoft has been preparing for BC from the start. If you look at a lot of the pre-scorpio news cycle in 2016 there were a lot of comments coming from MS about the end of console generations, and how Xbox is a family of devices. Crazy to think that it's been 4 years already. Based on those comments, they prepared for BC since at least then, and plenty more interviews and articles detail how they profiled games and collected data to ensure previous games would work on One X without issue, so they already had experience maintaining compatibility with different hardware. Contrast that with what Sony did with the launch of pro, where the system would down clock and disable CUs to maintain performance on par with Base PS4 running unpatched games. It took them months (a year maybe) to enable boost mode as an option, and it's basically offered without support. As in, if there is a game that has issues with it enabled, the supported resolution is to disable it. I don't actually know of any game that has issues running with boost mode, which makes it even stranger.
Regarding BC across multiple generations, Microsoft has the advantage of deep financial resources, plenty of skilled software engineers, and Direct X. And I don't think Direct X gets enough credit. It's a slightly higher level of abstraction, and it's MS controlled. It's easier to ensure that you get the desired output when you control the API as a whole. They also had the added motivation, because Xbox One went though a bit of a software drought, at least regarding exclusive software. BC allowed them to add value to their platform by adding games that were exclusive at least to the generation. You can't play Red Dead Redemption on PS4. Or Black Ops 1 and 2. And for a time, the only way to play Borderlands 1 on a current gen system was via BC. Not that it mattered to Microsoft, but it also added value to peoples back catalogs, because many of the games that went BC doubled in price on the used market. Sometimes they more than doubled, as I remember when Black Ops 2 and MW3 went backwards compatible, they went from being $3 games to being $15, and the demand was huge for them. It's crazy when games are that common but sell for significantly more that the average game on a platform. Then again, anyone look up how much Wii Sports is selling for right now?