First, the things that would not change. The Dreamcast would still fail and would sign Sega's exit from the hardware side of the market. XBox not being there does not change the fact that the PS2 is still around and that the Saturn was beaten soundly by its competition, so even if it gets one or two extra million sales, Dreamcast still fails. Likewise, the GameCube still sells about the same amount, maybe a couple million more, but again, the XBox did not hurt Nintendo's share of the market, the PS2 did that by itself. The PS2 would sell more or less the same amount, as at the point of 150+ million consoles, XBox not being there changes little.
After the 6th gen, however, things would change a lot. Nintendo would still go for an upgraded GC, which would be the Wii, and without any similar competition, Sony would take the PS3 gamble even harder. But in this case, the lack of a 360 is very important. At the time, the 360 was a clear and competitive alternative to the 599 US$ machine, especially to those who did not consider the Wii as a true competitor. But without a 360, they either go PS3 or go Wii. I imagine the PS3 would still have a terrible start, since even if there is little competition, lack of launch titles and exorbitant launch price makes it a difficult proposition. The Wii would still go on to be the best selling machine, but now it would gain a lot more attention from 3rd parties, as the alternative is a very expensive, very difficult to develop for machine (or PC).
Thus, 7th would not push graphics as much as it did, and neither would online gaming (both the Wii and the PS3 online services were comparatively inferior to those on the 360, though on the upside, the fact there was no paid online console could mean it does not become common going forward). Sony would start gaining momentum due to emergency measures with the PS3, but a lot more core games would have been developed with the Wii in mind (big games like GTA IV, Dead Space, Far Cry 3, CoD Modern Warfare or Mass Effect could have been Wii games, since the PS3 userbase alone may not have justified their development as PS3-PC games alone). With the Wii getting even more games, I imagine Sony would push their own motion control stuff even harder to try to keep up. I imagine the sales would be quite similar in the end, though the Wii getting more 3rd party support would mean its lifespan would be longer and wouldn't immediately dry up after thr 4th-5th year. Maybe it could have reached around 120m sales, while the PS3 would be around the 100m mark.
After that, the tables would turn, as Nintendo would be even more confident on the Wii strategy, and would go all in on the WiiU, though a longer Wii lifespan would make the WiiU release later, which could mean a console on par with the PS4 hardware-wise. Still, it would be expensive and confusing to those who initially bought the Wii, and would get a similar fate. It could have a much better launch due to the Wii having a bigger amount of core games than it had in reality, but it would still end up failing, with people going back in droves to the PS4. I imagine the WiiU could do N64 numbers, but not more than that, while the PS4 would be around 120-150m units. Perhaps at this point online would still be free, as there is no effective precedent for paid online, and Sony would not risk it to try to avoid sinking the PS brand now that they have it back on track. I'm not sure how midgen upgrades like the PS4 Pro would fare, but I imagine that, even if they released, it would not change much.
After that, the Switch and the PS5 would come out, and this would mark a full split of the market. The PS5 would stand as the single dedicated home console, while the Switch would be seen as a hybrid/dockable handheld. I imagine the numbers would be very similar to what we have right now, as the XBox XS is not really affecting the sales of either PS5 or Nintendo in any meaningful way. This would be the time where this home console monopoly would push many to just go PC, as I can imagine a Sony having this much control over the market would make them take controversial decisions which would cause people to abandon consoles altogether.