Choosing my frame rate on settings

My monitor is a Gigabyte M32U. It's an HDMI 2.1 display and can do 120Hz @4K with HDR 10 over HDMI or 144Hz @4K with HDR 10 on my gaming PC via the Display Port.

The issue with your monitor is that it's an older HDMI 2.0 display. This leads to a reduction in the available bandwidth. This is likely why you can't have both 120Hz and HDR.

I hate to disappoint you, but there isn't just one type of HDR. HDR 10 doesn't automatically mean your monitor has great HDR, it doesn't. What makes HDR good HDR is the displays maximum brightness. It is given in nits or one candelas per square meter (cd/m2). One nit is equal to one cd/m2. Typical brightness ratings range from 250 to 350 cd/m2 for monitors that perform general-purpose tasks. For good HDR, a brighter luminance rating such as 500 or more cd/m2 is desirable. Your display is an HDR 400 rated monitor.

Your HDR is just okay, not great. It won't really stand out though. On my M32U I'm not massively impressed with HDR so I've turned it off.

Could anyone with a new gen console, a 120Hz and a HDR10 screen device run a test in their on console to make sure the console can operate in 120Hz with HDR10 feature on and automatically detecting the type of screen?

My monitor works fine with both 120Hz and HDR,

I think some monitors, will only do one or the other, 60hz with HDR or 120hz without HDR, every monitor is different and you have to look around and do some research to know your getting the right monitor that's going to work with what your wanting it to do,

I'm using the LG 27GP950 a 4K 144Hz which can be overclocked to 160Hz and has HDR10 (HDR600), VRR, HDMI 2.1 and with the Series X also having HDMI 2.1 port, it all just works,

Some HDMI 2.0 monitors can have issues with Series X, but it depends on the monitor as each monitor is different from one another ( timings and features as such and so on ).

Monitors, specially ones that have been overclocked tend to use nonstandard timings which can cause issues with compatibility. Please make sure the monitor firmware is updat3d, and failing that you should call Acer about these compatibility issues. Xbox uses standard VESA resolutions and timings.

High frame rate and hight graphics at the same time. Hate the fact on some games on the xbox one x you have to choose between the two.

Personally I always choose frame rate

No, everything configures itself and the tv will automatically sync to the incoming signal