It was very true on their home consoles in the Gamecube and Wii U eras. I think this comes from a few reasons.
1. Nintendo first party games are always in extremely high demand on all of their consoles. Most people who buy Nintendo consoles buy Nintendo games, it's a tradition dating back to the NES where people buy a Nintendo console with 1-3 first party games, and maybe a third party game or two. Often the first party game includes killer apps, or the primary piece of software driving the console sales usually to the point where demand outstrips supply by a significant amount: (Major examples include: Super Mario Bros on NES, Tetris and Pokemon on GB, Wii Sports on Wii, Breath of the Wild, Mario Kart, and Animal Crossing on Switch).
2. N64, Gamecube, and Wii U had small numbers of third party games because of various reasons - mainly lower hardware sales.
3. While N64 had a small number of third party games, they were mostly exclusive. Gamecube and Wii U had very few third party exclusives, and often the multiplatforms (particularly on Gamecube) were inferior to those on other consoles: Gamecube controller was insufficient, there were fewer features, and often less content.
But there are other considerations. Nintendo's relationship with third parties is not the same as Sony or Microsoft. Technically, the top selling game on Gamecube is a third party game, but it's considered a first party game because it's published by Nintendo, made by an exclusively (or near exclusively) Nintendo developer, and involved numerous Nintendo licenses. Now, even if only part of that was true, usually those games get counted as first party games (or the colloquial "second party" game, even though technically the user/consumer is the second party). So, it's not really as black and white as it seems.
Scaling because of user base is a thing too. I don't even think the ratio of first party sales to third party sales differ a great deal between consoles (maybe 10-15% more third party on the better selling Nintendo consoles). A lot of it is simply, Wii sold 5 times as many consoles as Gamecube, and therefore sold about 5X as many first and third party games.
The highest selling unambiguously (or by all definitions a) third party game on Gamecube was Sonic 2, 1.73 million (and is the 14th best selling game on Gamecube). The highest on Wii is Just Dance 3 at 10 million (and is the 10th best selling game on Wii) - the numbers are fairly close to the 5:1 ratio for total hardware sales. Surprisingly, the highest selling third party game on Switch is Monster Hunter Rise at 7.7 million units and is the 23rd highest selling Switch game. This doesn't mean third party games are doing worse on Switch because Switch has literally 4 times as many games as the Wii, and nearly 20 times as many when compared to the Gamecube, take into account the points I said earlier about Nintendo owners usually buying first party games, and consider that third party software is fairly dilute on Switch compared to Gamecube and Wii.
I describe myself as a little dose of toxic masculinity.
