Honestly i find it pretty strange such rivalry even happened Sony and Nintendo fans normally go pretty well together.
I don't find it surprising in the least. I'd say MS/Nintendo get along better than MS/Sony or Nintendo/Sony.
Sony and Nintendo have more overlap than MS and Nintendo and a bloody history.
PSX versus N64, and the collapse of the Nintendo/Japanese third party home console. A lot of bad blood there.
Console versus handheld for Japanese games and consumers. A lot of people in the core fandom have a strong dislike for handhelds and a strong like of Japanese games. Others are confused that, say, 3DS doesn't get more support than it does.
DS/3DS versus PSP/Vita; Nintendo handheld dominance versus luxury, "gamers' dream" hardware. Doesn't help that Switch is a luxury handheld and that Vita failed.
I mean, you have the whole rivalry between Nintendo consoles and coreboxes, but, even there, that is mostly between Sony and Nintendo this generation, given the nature of the PS4. When people are angry that Nintendo still exists and does its own thing, they mostly want those games on PS4, not Xbox One, because PS4 is the market leader and there is more similarity in software profiles, when you look at the more niche side of PS4.
I mean, maybe it's just GAF, but coming from a predominately Nintendo oriented perspective on GAF, I don't really see Xbox come into the picture much. I do see PS a lot, and often there's tension there.
Which is perhaps weird because there's a lot of continuity in console profile between N64 and Xbox, but Nintendo really shed that identity with GCN and on.