tl;dr - You're looking at FPS performance that's tied to the GPU, not CPU. You're using the same GPU, so you get the same performance.
One thing you should be aware of is that the CPU only affects the speed/FPS when it is the limiting factor in Dolphin. If the CPU is the bottleneck, upgrading to a faster one will result in a boost to FPS. Often times, this happens when the game itself is demanding to run in Dolphin on any hardware (e.g. Rogue Squadron 3) or the given hardware is too weak to run Dolphin adequately (e.g. using a low-clock ultra-low-voltage CPU for most games in Dolphin).
On the other side, the GPU only affects the speed/FPS when it is the limiting factor in Dolphin. Many times these days, this happens when trying to raise the graphical output from Dolphin beyond what the GPU can handle at fullspeed, i.e. raising the internal resolution too high or adding demanding amounts of AA.
So, to understand what's happening here, I'd say you were never bottlenecked by your old CPU, but your GPU. For Dolphin, you can basically think of being bottlenecked by the CPU by answering this question. "If I run the game at 1x internal resolution, no AA, no enhancements, does it run at fullspeed?"
If no, then your CPU is likely the bottleneck*, since even most integrated GPUs handle 1x IR without issue, and the fault will lie with another component.
If yes, your CPU isn't going to bottleneck your system when raising the resolution; your GPU, however, will then become the bottleneck as you add more demanding features for it to handle.
If you really want to see where the Ryzen beats your old FX, test both machines with Dolphin at 1x IR, no enhancements, and disable any frame rate/speed limits. The Ryzen will run laps around the FX in terms of FPS. To grossly simply a bunch of technical stuff, that's where the bulk of Dolphin's CPU workload goes towards, getting the game up and running as long as the GPU doesn't slow things down. After that, most slowdowns originate from the GPU's workload, depending on what it's tasked with handling.
* There may be cases where both the CPU and GPU are too weak to run Dolphin fullspeed even at 1x IR. In that case, it's harder to distinguish which one is at fault.
One thing you should be aware of is that the CPU only affects the speed/FPS when it is the limiting factor in Dolphin. If the CPU is the bottleneck, upgrading to a faster one will result in a boost to FPS. Often times, this happens when the game itself is demanding to run in Dolphin on any hardware (e.g. Rogue Squadron 3) or the given hardware is too weak to run Dolphin adequately (e.g. using a low-clock ultra-low-voltage CPU for most games in Dolphin).
On the other side, the GPU only affects the speed/FPS when it is the limiting factor in Dolphin. Many times these days, this happens when trying to raise the graphical output from Dolphin beyond what the GPU can handle at fullspeed, i.e. raising the internal resolution too high or adding demanding amounts of AA.
So, to understand what's happening here, I'd say you were never bottlenecked by your old CPU, but your GPU. For Dolphin, you can basically think of being bottlenecked by the CPU by answering this question. "If I run the game at 1x internal resolution, no AA, no enhancements, does it run at fullspeed?"
If no, then your CPU is likely the bottleneck*, since even most integrated GPUs handle 1x IR without issue, and the fault will lie with another component.
If yes, your CPU isn't going to bottleneck your system when raising the resolution; your GPU, however, will then become the bottleneck as you add more demanding features for it to handle.
If you really want to see where the Ryzen beats your old FX, test both machines with Dolphin at 1x IR, no enhancements, and disable any frame rate/speed limits. The Ryzen will run laps around the FX in terms of FPS. To grossly simply a bunch of technical stuff, that's where the bulk of Dolphin's CPU workload goes towards, getting the game up and running as long as the GPU doesn't slow things down. After that, most slowdowns originate from the GPU's workload, depending on what it's tasked with handling.
* There may be cases where both the CPU and GPU are too weak to run Dolphin fullspeed even at 1x IR. In that case, it's harder to distinguish which one is at fault.