Navi will be more efficient, giving more graphical power for the same number of watts. But Vega will apparently continue as AMD's flagship gaming card, with Navi launching as a medium tier. (I don't know specifically why that is, but multiple sources say so.) Microsoft want to be undisputed performance leader, so that might make Vega attractive to them. Also, Vega features effectual FP64 calculation, which is rare for gaming but useful for machine learning AI and other advanced tasks. And Microsoft have officially said they want to use the same hardware in the next Xbox and their gaming servers.
It's important to note that this imagined scenario where Sony use Navi and Microsoft use Vega wouldn't have an inevitable winner. It could go all sorts of ways. Maybe Microsoft does get the highest TF card, and their Xbox One X experience lets them cool it effectively without much extra cost. There's rumors that Navi isn't as effective or power-thrifty as meant to be, and maybe PS5 could end up needing extra cooling redesign and downclocks at the last minute, which make it just as expensive as Xbox but notably less powerful.
Or, maybe Vega has higher TF but a lack of efficiency, and the usage of chip space for new server-specific features makes the next Xbox less performant than expected. Meanwhile, a careful Navi design with smartly customized gaming-relevant features allows PS5 to produce graphics basically indistinguishable from Anaconda, but at a much lower price.
In other words, neither choice is necessarily bad. Both have potential drawbacks, but both also have potential advantages for what the different platforms hope to accomplish. And no matter what they design for, some tech bets simply don't pan out.