I think Gemini is wrong. The Pi5 appears to output RGB(16-235). On the Zidoo, selecting "Priority RGB444 8BIT for non-4K content" and "HDMI Range 16-235" (important: do not let it on auto!) produces a similar result. All the other options were worse in some way. With these settings, the 3D on my Zidoo is on par with the Pi5, or at least very close. What results do you get when you compare these settings?
Note, this should all just work automatically if all the components in the chain do the right thing.
The EDID from the sink (display) reports the list of HDMI modes it supports.
These include CE/CEA modes (TV-style timings) and IT/DMT modes (monitor-style timings).
There are different default rules for RGB quantisation range: CE/CEA video timings generally use limited range, while IT/DMT timings generally use full range (with some exceptions).
When the source (Pi) outputs HDMI, it can indicate the RGB quantisation range in the AVI InfoFrame. The sink should use that signalling, or the appropriate default for the video format, to interpret the pixel values correctly.
So everything should just work by default. However, it is not uncommon for devices in the chain to get this wrong - for example, a sink might not correctly respect the InfoFrame signalling, or a source might choose or signal the wrong limited/full-range setting. This results in things such as blacks being crushed or looking grey, and whites being clipped.
Colourspace/colourimetry (for example BT.601 vs BT.709) is another similar setting. The source signals the colour encoding in the AVI InfoFrame, and the sink is expected to interpret the pixels accordingly. Again, this should normally work without any manual settings, but interoperability bugs or incorrect signalling can result in incorrect colours.