Sounds like it is a charging battery connection or you would not see any volts. Did it have a barrel type battery attached to the board originally?
If it did then you need 3.7v of battery as those barrels were only made in 3.6-3.7v AFIAK. If it never had a barrel then it would have used a 3.6v Nicad battery pack. Yes you are overcharging the two Eneloops by a lot ! I would use three of them.

Horun wrote on 2020-02-15, 17:16:

Sounds like it is a charging battery connection or you would not see any volts. Did it have a barrel type battery attached to the board originally?
If it did then you need 3.7v of battery as those barrels were only made in 3.6-3.7v AFIAK. If it never had a barrel then it would have used a 3.6v Nicad battery pack. Yes you are overcharging the two Eneloops by a lot ! I would use three of them.

It did not have any battery attached at all and doesn't seem like it every did. The motherboard manual doesn't mention it and the board doesn't have placeholder for it. Unfortunately the motherboard doesn't specify if it's a charging port.

I was told that you need to measure current when a battery is connected and the computer is turned on and that will tell if it will charge it or not.

Baoran wrote on 2020-02-15, 17:50:

I was told that you need to measure current when a battery is connected and the computer is turned on and that will tell if it will charge it or not.

The problem with that is most CMOS chips only need about a dozen microamps to keep it's memory alive, and it does not account for the reverse charge current going into battery when on. Better would be measure the current with system on going to the battery, if only a few microamps then not charging as it would require something in the 10 milliamp range or more to charge. There is a Vogans topic on this but can't find it...

I tried to measure the current both when turned on and off and didn't detect anything with my multimeter. I'm attaching a photo of the header, next to it are two diodes (I'm guessing) connected in series to the 3.6v pin. Maybe one of you can deduce something from it?

Also, noticed the rechargeable batteries are not getting warm when the computer is running. Does it mean they're not charging, typically they would feel warm when plugged to a regular wall charger.

Ok, CR2 definately makes sure the battery is connected properly or else no volts to cmos, also would prevent any charging. You need a 4.5v +/- battery source (4.5v- 0.7v for diode=3.8v) so three AAA or AA batteries is what it needs. It probably had something like the attached picture originally which are near impossible to find.