Just some clarifying comments ...

disclaimer - I've never had to do anything on my X5's AC, so this is just from general understanding of AC systems and maintaining my other cars. But I think the R12 or R134a and physics still apply.

On the manifold gauge set, first thing to do before hooking it up is to close both of the valves at the manifold (the brass thing that the gauges and lines connect to). Turn the red and blue knobs right next to the gauges clockwise to close them. With those valves closed, the gauges will still read the pressure in the line, but nothing else will flow - e.g., from the yellow fill line to either side, or from blue to red.

If those valves are both open, the H and L pressures will be equalized within the manifold, which is definitely not what you want. If either one is open, it will connect that line to the output (center, yellow) line, which appears to be plugged in the photo.

Then hook up the lines at the ports. From your description, it sounds like that is done exactly right. You will also want to bleed air from the lines, but that is a secondary factor at this point.

With then engine off and the AC system stabilized (i.e., given enough time after shutdown to allow things to flow and equalize), you should be reading equal pressures in H and L. Looks good.

The pressure you read in this state is mostly an indication that you have (1) gauges hooked up right, (2) freon in the system, and (3) ambient temperature (not the thermometer on the wall in your garage, but the effective temp of the engine bay, specifically the entire AC system). If you are very low on freon, there will be no liquid in the system, and the pressure will be below whatever you would get when you do have liquid in the system. Details follow.

When you do have liquid in the system, it is a saturated vapor, and whether it is overfilled, underfilled, or perfect, the pressure reading will vary the same as ambient temperature goes up and down.

From the numbers you're reporting, if the gauges are hooked up right and your numbers are accurate, it sounds like you are low on freon and do not have a saturated vapor. It is all vapor. No liquid in there. This would prevent the compressor from running.

If you look at the gauges, the red numbers (temperature units) on inner rings are refrigerant dependant - specified as R134a for your gauges, and they assume that the refrigerant is a saturated vapor. In a saturated vapor, changes in volume at constant temperature cause it to condense/evaporate fast enough that the pressure remains constant. So there is a direct relationship between temperature and pressure IF it is a saturated vapor (at the very least, this means that both liquid and gas are present). So these markings on the gauges are doing this pressure-to-temperature lookup for you - valid for a saturated vapor.

Since your pressure numbers are well below what you'd expect at your temp, it sounds like you have only gas in there. Freon needed.

Then, you reported that after running the AC, pressures in both sides increased equally from 31 to 37.
- equal could be explained by the manifold valves mistakenly opened. hopefully not.
- equal and low means the compressor is not compressing. probably.
- the raise in temperature is probably due simply to the ambient temperature rising. Even with your AC belt removed (as an extreme example), running the engine will warm everything up in the engine bay, including the AC lines. This will cause the pressure readings to increase. As the temp rises, those gauge readings will keep going up.

Maybe all that was done correctly already, and was clear to everyone. But if not, hopefully it helps.

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2001 X5 3.0i, 203k miles, AT, owned since 2014