The UGreen Revodok MAX 208 (on Green web site) indicates "Dual 4K@60Hz Display: Connect dual displays through Thunderbolt 4 for smoother multitasking and more screen space." It does not mention that two monitors with 1920x1080 would work with 120 Hz, however. It does say that one monitor @ 2K works with 144 Hz.
As for the Anker HDMI Cable, from what I can see here https://www.anker.com/uk/products/a87e0-anker-hdmi-cable-usbc-to-hdmi-cord?variant=57823413338486 it specifically mentions 60 Hz but not 120 Hz.
Technische Daten
Compatibility
Dell XPS 13, Dell XPS 15, Galaxy S23, Galaxy S24, MacBook Air, iPad Air, iPad Pro, macbook pro
Connector type
HDMI, USB Type C
Data Transfer Rate
60 Hz
Number of Pins
43
Maximum Voltage
10 Volts (DC)
Material
Nylon-Braided
Length
3 ft
I'm not criticizing your setup but I would suggest you focus on the cables/dock because the behaviors you describe sound more like a cable or dock issue that something with the Mac itself. And you said it is the HDMI one that is ignored.
Apple says ( How many displays can be connected to MacBook Air - Apple Support )
Two external displays
Supports two external displays in the following configuration:
Two displays up to a native resolution of 6K (6144 x 3456) at 60Hz or 4K (3840 x 2160) at 144Hz
You indicated both monitors are at 1920x1080 120Hz, I would think that would work but the above specifies 144Hz, should it also work at 120 Hz?
But as I said, I don't think it is the Mac that is the problem. Can you try running this stuff at 60 Hz (as a test)?