You must have a fully-calibrated setup for this to even vaguely be possible.
No calibration, no chance.
OK, this is going to be a vague answer because the data needed to correctly assess it isn't available.
More detail lower, I actually ran a full calibration test to prove this can be done.
You are dealing with colour handling in a situation that is not colour-managed. I doubt you will improve this without hardware/profiling calibration at both ends of your connection & even then is doubtful.
RDC is sending the colour of the actual pixel as perceived at source, not the colour as managed by either your calibration profile nor the one at the remote machine. The receiver may or may not be interpreting this through its own profile… I would imagine not; it will be leaving that to whatever regular desktop profile you are using, which is unlikely to be accurately calibrated anyway.
If your digital colour meter has variable interpretations, try sRGB & Native. sRGB will be the 'generic' profile the machines think they are using, but requires two conversions, from managed pallet to native, then back to sRGB. Native will report the colour the computer actually "thinks" it is, without any interpretation.
The more accurate your calibration is, the more generations of this two-way sRGB translation you can go through before you start to see colour-shift.
Essentially, it's a scenario never designed to be that accurate; it's a business solution to transmit usable information so one machine can remotely manage another, not an art solution one would use for accurate colour workflow.
If neither screen has been accurately calibrated, then you literally have no clue as to which, if either, is actually the correct colour, nor which, if any, translations have been used between source & recipient.
TL:DR - colour workflow management is one of the most complex structures a computer user has to handle. It is simple [& virtually invisible] once set up, but blindingly confusing to the average user - entire books have been written about it. If you don't have accurate hardware calibration in the first place - which 99 out of 100 computers don't have - then you either have a massive learning curve to go through, or you just accept it for what it is, unmanaged colour… no two people will ever see the same thing.
I decided to investigate this more thoroughly, to see if it in fact could be done with any accuracy over RDC on a fully-calibrated system. The results were far better than I expected.
Native vs sRGB values - on a fully managed system. Images in sRGB.
You can also use these to test your browser's compliance - the blue area being measured should be identical on all 6 areas you can see it PLUS should match the colour of the real 'Add Answer' button on this page. If they don't, your calibration is out.
Before I did this next step, I tried the same experiment before calibrating the PC. The results were absolutely terrible - I've simply never cared before because I remote to this PC for Windows testing, not for artwork. I didn't take pictures - perhaps I should have done.
Having just calibrated a Win10 PC [roughly, low quality, fast setting] I have then connected to it over RDC, opened this web page in Chrome & put the Digital Colour Meter over that RDC web page - so this image is a composite.
To the left; Chrome on Win 10 over RDC, showing the sRGB image already posted above; to the right, the current 'live' colour meter on this local machine, measuring the RDC-transmitted image.
It ain't perfect, but it's close [a full calibration would be closer but that would take me an hour] This has now been through maybe 6 different colour-mappings to get thus far, so being only 1 out through that many translations I think is pretty damn good!