It probably wouldn't have been "Nintendo 3" though. I mean, it's the Playstation 3, not the Sony 3
Would NES3 sound so bad though? Although to be honest if they were going to use a numerical naming system I think the Gamecube would have been the best place to start that. I mean, who wouldn't want a Gamecube 2, 3, 4 etc.?
The Gamecube was a failure though, they probably didn't want to be associated with that. They tried to stick to the Wii-branding a bit. Maybe Switch can be the starting point.
It doesn't have always to be numbers either. The NES was followed by the Super NES (or Super Famicom). I think that is also pretty clear. You have some steps, like Mega, Ultra and so on too. As I am a technical nerd, I would go with SI-prefixes. The Mega Switch. Followed by the Giga Switch. Followed by the Tera Switch!
Dreamcast by the way is the best console name ever.
And yeah, what Microsoft does is just confusing. Well, they stick with their brand Xbox, but it is very difficult to guess the order if you only hear the names. The most likely order would be: Xbox One, Xbox, Xbox 360. And Series X fits nowhere.