Looking at the linked board manual reveals that it doesn't support any 'turbo' switch. So this is simply a case having it, while there is no purpose - maybe except for the fun of having a button that switches the illuminated segments :)
It might be safe to assume that the board has been updated or the case been used regardless. After all, why throw away a perfect good case just because it has some unused feature?
Looks like a typical home built machine anyway.
Yet the PC's case has an LED module that is configured to show the CPU's current speed setting
Is it really? Or is it just displaying some number that has been jumpered? Have you looked inside how it's wired to the board - also, what board it is?
Notice that the LED module comprises of three digits, meaning that the LED module was meant to be used on systems with over 100MHz.
It seems like a 2½ digit display, so 199 should be the maximum value that can be displayed, shouldn't it?
The number of digits in use doesn't say anything about the board. Not even about the time it was manufactured. Using a 2½ digit display doesn't cost much more than a two digit one, but allows setting nice fancy numbers. The machine does look like it's self (or small shop) assembled, so they used what was available. Think of it like any other form of 'modding' just meant to support the ego of its owner.
Bottom line, the display used doesn't say anything about the time or the board.
I thought the whole point of the turbo switch was to enable compatibility with games that expected 4.77MHz. What was the purpose of the slow/fast turbo setting on PCs whose CPUs could not slow down to 4.77MHz?
It's important to keep in mind that a turbo switch is a feature of the case, not of the board. So what function it may have, if any, is defined by the board; you may want to look up the board documentation and check the wiring used.
Besides that, it was in general a way to slow down - though not necessarily to 4.77 MHz.
Throttling down to 4.77 MHz was only a thing for 8088 based (aka PC/XT) machines. Already with 80286 based computers (starting in 1984) this no longer made any sense. Here, if at all, a reduction to 6 or 8 MHz, like the base AT or later models, was used. With introduction of the first 80386 in 1987 the relation changed again. A 386 running at the same speed as a 286 will perform considerably faster, so reducing the clock speed would not bring it down to the same level as an AT.
At the time most motherboards supporting turbo modes (or rather reduced speed modes) simply disabled the on board cache to reduce performance.
The discrepancy increased with 486 and Pentium even more. A Pentium running at 4.77 MHz is several times faster than an genuine 8088 PC at 4.77 MHz. At the time this machine was contemporary (late '90s), a turbo switch had for most part lost any meaning. For one, there weren't many games still able to run on machines that new at all - and even fewer people who wanted to run the old stuff.
In the late '90s, games had to comply with a range of 25 MHz 386 to 200 MHz Pentium, so none were still written to rely solely on clock speed.