You should check your 3rd party programs configuration first, like color space (PC uses 0-255 while TV or youtube and some programs by default expect 16-255 if I recall), and color format.
I believe you can make your GPU send a different signal in Radeon Settings, too. Play around with these and see if it changes anything.
You said yourself that the card software displays the image correctly, suggesting that it's the 3rd party programs misconfiguration.
The color space has been notified and set to be 0-255 as the main signal is RGB. But still there is distortion.
A big problem is that I don't know the total color process chain in this kid of case. Why the GPU is capable of affecting on the encoded output video color?
Can you connect a different PC or laptop to see if the issue persists?
Can you check if the color settings on source PC and target capture card are the same? Like 16/24/32 bit color etc.
Have you tried a different cable?
This whole issue is very weird. Since the card software shows the image correctly, there should be no issues at all. Why do you suspect the graphics card to be the culprit?