If you look at the average lifetime of a (successful) console, you're looking at between 5 to 10 years. Within that time, multiple revisions take place, often to replace peripheral components to reduce costs, but in the case of the longest living consoles they will often get a design refresh, including a redesign of the CPU / GPU. I can think of only one example in recent memory where a console switched manufacturers for these core components during its lifetime (that example being the GPU for the Xbox 360, which switched from AMD to IBM). In the case of the Xbox 360, IBM were already manufacturing the CPU, and the redesign combined the CPU and GPU into a single SoC. Aside from that one example, there haven't (to my knowledge) been any other examples of companies switching manufacturers for their core components mid cycle. Contrast this with the PC business, where OEMs will frequently swap between motherboard/CPU/GPU manufacturers, and you'd have to wonder what makes the dynamics in the console market different.
I spent a while in the industry(and shipped some god-awful titles) so I tend to trust the people I know(and don't plan on outing them) but can understand if you don't want to take the word of a random person on the internet.
I know you were probably talking about traditional consoles(xbox/ps/etc) but when you start including portables(with the switch just smashing the 1 year mark at 15M units) then the field opens up significantly. You've got at least 5-6 different GPU vendors and a whole host of companies providing SoCs. Embedded GPUs have been making huge strides and I wouldn't be surprised to see some of the top end of that space start nipping at NVidia/AMD soon. Lots of the people in that domain came from desktop(Adreno is a anagram of Radeon for instance ;)).
When you're talking 15M+ units being able to drop $1-2 from BOM is a huge motivator for things like this. When you get into the millions of dollars worth of savings you can fund a lot of engineering time.
They also may never go through with the switch but use it as leverage for a better deal. Back when I did SoC evals we'd take products right up to the brink of production just to apply pressure. It was almost like a game of chicken between 2-3 vendors to see who blinked first and gave us a better BoM/deal on the core chip. These weren't easy bring-ups but usually were vastly different GPUs + CPU combos. It's very much a thing that happens, at least back when I was involved with stuff like that ~6 years ago.