It looks as if there should be a diode (D4) there to prevent charging and it's missing. A lot of possibilities:
-This version of the motherboard was never meant to be used with a non-rechargeable battery, but the assembler didn't care/know
-The missing components are a manufacturing error
-Someone changed the original battery (a soldering iron has touched the board, see the IC pin between PCB marking C9 and BT1)
-The diode is not missing, but it's D3 and it's broken, very easy to measure
You can't really know, but you are right in this: a non-rechargeable battery shouldn't be there

Thanks for the reply konc! I believe this is the original battery because its manufacture date is 1993, so maybe D3 has just died as you said.. I dont think it would have lasted this long if it was a manufacturing defect or missing component, would it? Wouldnt it have killed the battery pretty quickly if it was trying to charge a non-rechargeable battery?

Another question: I have these green NiMh rechargable batteries which fit perfectly between the "A" and "B" contacts. I measured the voltage here as well, and its also getting +5v. Can I use these to replace the battery and assume the board will keep it charged?

As you've noticed, the charger is little more than a diode and resistor connecting it in parallel with the 5V line, and that will be what determines if a NiMH will last decently or overheat when full on such a basic NiCD charger → https://batteryuniversity.com/article/bu-408- … l-metal-hydride
In other words you may (or may not) have to increase the resistor to further slow down the charge!

And yeah, can't say I've ever seen a PC with a lithium thionil battery, in the mainstream that's an Apple thing 😀
At least it wasn't a Maxell 😁 https://68kmla.org/bb/index.php?threads/warni … atteries.25169/