I really wouldn't even say Sony has been super successful in this. Back in the N64 vs PS1 days, PS1 was viewed as for an older audience when compared to Nintendo. Nintendo seems to be the only one who has successfully been accepted as a family friendly gaming system, and I think that is because they take a similar approach as Disney does with many of the films: they try to make their product fun for the entire family, not just a targeted audience.
Third parties really are where it is at for truly kids focused games. Unfortunately, most of those are garbage games. Remember all those terrible THQ Nickelodeon games? The Lego based games have been great and once again are great for entire families to play together.
Yes, the Third Party kiddie games were often bad (not all, but often) ...but the thing was that PS1 was the
everyone console. Sony worked the family audience as well as every other demographic. So Sony didn't have to make the family audience come with their own games, they just had to steer the entire market their way.
Many homes with kids had N64s, that was a lot of its identity, but the inexpensiveness and the proliferation of games from popular brands made PS1 an extremely common additional platform, if not the only box in a home. The Nickelodeon games, the Chocobo Racers, the Backyard Sports, the Mary-Kate and Ashley games, the Dragon Tales games, the Bob the Builder games, tons of Disney games, a couple of Smurfs games and some early Ubi stuff, tons of just genre stuff like racers or platformers or puzzlers or sims that players of any age could pick up and toy around with...
Everything you could want was on PS1. Then Sony supplemented it with their own brands of platformers and family-friendly games like Spyro and Crash. (It also produced some licensed Disney stuff itself.) Those did help establish the Sony brand as all-audiences from First Party on down, but kid gamers were there because the whole market was there, whether Sony tried or not.
To some extent, the PS2 was that "everyone machine" as well, although it oddly had less appeal in the young market (or maybe just my impression of "kiddie" games against the aggro culture of the time didn't consider it in the same context?) Game development prices had gone up, all-ages game design and access level scalability had solidified si that kids' games weren't just kids' games, there were sports games which were so colorful and amped-up that anybody could enjoy them, and the base level of target demographic just aged up for like Jak 2 or Kingdom Hearts. I don't think Sony lost the market with this change of target demographics because I don't think Gamecube really won the kiddie gamers either - maybe they just fell back to GB/GBA and eventually mobile? that audience practically disappeared from consoles by the PS360 era and only exists now because consoles are cheaper than PCs to play Minecraft or Roblox on - but there doesn't seem to be as much age range in the PS2 library even though it was an even more dominant platform.
MS for sure never was able to capture kid gamers, but I think the shift in audience I'm describing is a bit of what happened. Big ol' Xbox OG was never going to get kids (despite having breakaway controller cords...what a sheer blessing this could have been for parents!), but that wad ex0ected and they only tried as hard as would be expected of s playform holder at the time. They did try with Kinect to some success (on the heels of Sony's own kid/family EyeToy device, which was a success especially in Europe yet struggled some the next gen around the same time Kinect was ebbing and flowing,) but the writing was on the wall I think by that point. Little kids had a growing number of other ways of occupying their time.
MS still made all-audience games and products when it could, but by the time they had entered the console market with even the first Xbox, it was already close to too late to invite the youngest players in.