Wasn't the first AMD 2x4c design CPU called Interlagos (server 16c Bulldozer CPU) introduced in 2011? PS4 console is 2013 product (and PS4 CPU core Bobcat is derived from Bulldozer). It just looks to me that consoles and custom business is based on other AMD products not the other way IMHO. And it makes sence that custom department is using already developed products of CPU and GPU departments and combines them together into custom chip.
PS4 doesn't have a Bobcat, but a Jaguar, and Bobcat is not derived from Bulldozer. Bobcat was a second design effort made at the same time by a different team (AMD Austin) than the Bulldozer cores. It was a very bare-bones done-on-the-cheap design sort of meant for the cheap netbook/third world markets, built by a tiny team in a short time with little engineering resources and designed for the TSMC 40nm process that was inferior to the 32nm SOI the BD cores were built for. It was much more successful than expected; it was much faster than expected for such a bare-bones design and notably A0 silicon had no showstopping bugs and shipped to customers, something almost unheard of in CPU design.
It's unexpected success at a time AMD was otherwise doing very badly was also sort of the immediate cause for why it went wrong, as AMD's stock options turned worthless and they could not afford to compensate the second design team enough, and Samsung first poached the lead designer (Brad Burgess), who then proceeded to pick the best people who previously worked under him at AMD and formed the core of the Samsung Austin Research Center (SARC) from them. SARC would go on to design and implement the Samsung Exynos M* Arm cores.
This was really bad for the immediate future and plans of the core, as AMD mostly had interns and unqualified lifers left at the team, and follow-on cores lagged behind schedules. Also, soon after Jaguar finally shipped, Intel finally got Atom right with Silvermont, which they were also willing to flood to the market not just at cost but at a
negative price (Intel sold Silvermont + network chips for less than what the network chips cost alone, so long as the Silvermonts were not scrapped but had to be sold as products. This was meant to help Silvermont push into the android tablet market, and maybe later into phones, but no-one actually wanted x86 android at the time, and so it just crashed the netbook market.)
Being synthesisable for the TSMC processes meant that the Bobcat/Jaguar design was in the right place when the console refreshes came around, and AMD was probably saved from bankruptcy by being able to sell complete solutions to the console makers, winning both the big bids.
As a final note, the cat cores basically never borrowed anything from the construction cores, but the reverse is not true. The cat cores were the first to implement the perceptron branch predictor, and idea that was bounced around in academia for a while before that. It was probably chosen for the cat cores because the existing predictors were too big and power-hungry for such a small core, and if they had to implement a new one with limited resources, they wanted to pick one that was conceptually simple and easy to implement, which the perceptron predictor is. It was among the things that worked better than expected -- compared to the finely tuned large monstrosities found in larger mainline cores, the predictor is supposed to be quite stupid, but is also very fast, uses little power and is tiny. In practice, it actually often outperformed the predictor in BD, and was thus introduced into the Piledriver, and is still the first stage of branch prediction in Zen 2.