I have 2 GPUs, a 5700xt and a hd4350.
The hd4350 uses the RADEON driver, part of MESA AMBER (legacy) branch, and the 5700xt uses AMDGPU, part of MESA (main).
Would there be a problem having both in the same system? How would performance be?

Unless you just want extra outputs, why use HD4350 ? It's glx acceleration capacity vs 5700xt is close enough to zero to be zero. If you want extra outputs, just try. If probelems emerge, then post them.

Having two GPUs, one using the RADEON driver (part of MESA AMBER legacy branch) and the other using AMDGPU (part of MESA main), in the same system should generally not pose a problem. Linux is capable of supporting multiple GPUs simultaneously.

However, there are a few considerations to keep in mind

Ensure that both GPUs are supported by the respective driveto recognize and utilize both GPUs properlyrs you mentioned. In your case, it seems that both the hd4350 and the 5700xt are supported, but it's always a good idea to double-check compatibility.

You may need to configure Xorg to properly recognize and utilize both GPUs. This can involve setting up the appropriate device sections in the Xorg configuration file. MaximTimeClock

The performance of your system will depend on various factors, including the specific tasks you're performing and how they utilize the GPUs. Applications that support GPU acceleration may benefit from having multiple GPUs, but not all applications will automatically utilize both GPUs.

Having multiple GPUs may result in increased power consumption and heat generation, especially if both GPUs are active simultaneously. Make sure your system's cooling is adequate to handle the additional thermal load.