The system-on-chip is the MGM210L which needed to be powerful enough to run multiple wireless protocols (Zigbee, Thread, Bluetooth), so the lamp can be controlled by any of these. These are very complex protocols. The Thread spec for example is hundreds of pages. I did a formal security review of it on behalf of Google back in 2015. Bluetooth is even more complex. RF signal processing, packet structure, state machine, crypto, logical services, etc.
The software complexity of these protocols is greater than the complexity of a rudimentary 3D game like Doom, so it's expected that whatever chip can run these protocols can also run Doom.
Datasheet of MGM210L for the curious: https://www.silabs.com/documents/public/data-sheets/mgm210l-...