(When selecting an answer, use Stephen's - I just put this here because it is too much text for a comment)
Most Important: It's a Game of Raw Numbers.
We can ignore more complicated stuff and simply go for bandwidth:
ISA had at the time a bandwidth around 16 MiB/s (*2)
VLB (*1) and PCI offered up to 133 MiB/s bandwidth.
A 320 x 200 screen needs 64 kB (lets say 64 KiB) (*3)
With 30 fps, which can be considered good (*4), that adds up to ~2 MiB/s
2 MiB/s is 17% of the raw ISA bandwidth
but it's only 2 % of VLB and PCI's bandwidth
That is worlds for a game like DOOM, that did in the days shined due its many ways to make 3D run on barely able hardware. Often forgotten today, Doom not only reduced detail, but as well screen size. It offered 9 sizes, from 320x200 down to 96x48 (*5), and having a letterbox around the action was common, not the exception.
Performance One Can Feel
Gaining 14% additional CPU time to operate is nothing for an application, but huge on a resource strapped game. It's the difference of running DOOM in at least one screen mode up.
And between an ISA system capable to use 320x200 and the same CPU setup with PCI of at least doubling the frame rate - that's from barely playable to lightning fast.
Either effect is quite visible. All without changing anything within the game.
Downward Compatibility is the Name of the Game
PCI (and VLB) were made to have (existing) I/O hardware appear to the system like before. VLB a bit more than PCI, as PCI. In either case VGA memory and control was mapped at the same addresses than on ISA. A worthwhile effort as all software would benefit right away without any modification. It would need no knowledge about the changed bus system - the same way a 8086 software does not need to know if it's running on a Pentium.
So it was downward compatibility of the new bus systems - made by its developers to be able to sell PCI systems. After all, who would buy an incompatible PC, one were the Flight Simulator doesn't work like on an 8088?
But how is this managed "automagically". How is this forward/upward compatibility implemented?
There is nothing magic. The VGA memory was with PCI available at the same real address as with ISA before: A0000h. No change in the (visible) HSI (*6), so no need to change the software.
Your [@StephenKitt's] improved answer implies to me that original VGA was designed with upward compatibility to a 32-bit system bus.
No, if at all, the other way around. The newer busses were the same, except faster and wider. And both changes are invisible to software. From a software PoV there is no difference on how fast a CPU runs and/or if an access is done in bytes or words. That's all covered by the CPU hardware. The only measurable difference is speed. Something software will only 'feel' in performance (which is what your question is about), but not in operation.
Also that faster busses are transparent because slower busses just introduce wait-states.
No, there is no relation between software and bus (or CPU) speed. That is, no other than over all performance. Software does not need to be rewritten for different clocked CPU's, bus width or speeds. It will just show in what it can crank thru in course of time. In case of your PCI/ISA question, DOOM is simply able to transfer a frame in less time, thus either producing more frames in a given time (aka fps), or frames with a higher resolution (selected in the option screen). Either way: Better Gaming.
Lesser but Related: Pentium and VLB Sucked.
I do not remember the full parts, but AFAIR Pentium chipsets and the VLB protocol did not go well with each other resulting in a meagre performance, making a Pentium-VLB slower than a 486-VLB, while Pentium-PCI did work well. This not only made VLB obsolete soon, but also added a lot of less than favourable DOOM results.
Carmack and Romero, the DOOM guys, did BTW think that VLB was the way of the future to go and PCI just a strange side development by Intel, as one of them stated at the time.
*1 - Never forget VLB, it was the go to for gamer at the time)
*2 - Well, ISA had no fixed defined bandwith, as it always depended on the system. Early PC managed 2 MiB/s, while original AT did 6 MiB, later AT 8 MiB. Bus clock was ties to CPU clock, so faster AT clones could apply 10..12 MHz, neting up to 12 MiB/s. Many I/O cards would comply. With later 386 passing 16 MHz CPU clock, Chipsets started to handle I/O clock as a integer, non binary fraction of CPU clock. Like 33 MHz Systems offering 6.66 MHz, 8.33, 11 or 16MHz. Again later almost arbitrary dividers were offered.
At the time of DOOM, most ISA cards worked fine at 12 MHz or above. This already brought a huge speed advantage and higher frame rates, all without VLB/PCI.
For the above calculation I use 16 MiB/s as it represents what a good ISA card could deliver when plugged into a state of the art 386/486 system. After all, while using 8 MiB/s might show an even higher speed up, I do not believe it be fair to compare the new PCI bus system with anything else than the top end of its predecessor.
*3 - 320x200 is full screen and was almost never played. Highest regular mode was 320x168 with the well known status bar at the bottom of the screen.Saving these 32 lines brought already a noticeable difference in frame rate, even on high end systems.
*4 - Looking back shows how much pampered we're today. Playing Doom at 20-30 fps was great.
*5 - Yes, way less than a Game Boy - not sure if anyone ever successfully played in that mode.
*6 - Hardware Software Interface - a fancy name for addresses by software and their semantics.