My 144hz 1920 x 1080p monitor does not support my Xbox Series S while trying to run 120hz ?

for 120hz you need i think hdmi2.1.

besides connecting the s at 120hz is not going to magically let it run at 120fps, you will be lucky to get 40fps on most games. kills my eyes running so slow.

HDMI 2.1 is only needed if you plan to game at 4K for 120hz , on a 4K 144hz monitor, However there's no gaming monitors available at the moment, that yet has HDMI 2.1, the first HDMI 2.1 monitors that are likely to come in are around the same time as the
Series X (Q4) launch or early 2021 (Q1),

1080p and some 1440p monitors can do 120hz over HDMI 1.4 if it's supported, but best ideally is HDMI 2.0 for 1440p for 120hz,

HDMI 2.0 can do 1080p / 1440p at 120hz and 4K at 60hz,

HDMI 2.1 can do 1080p / 1440p / 4K at 120hz.

For the Xbox Series X, it best to get a 4K TV or monitor that supports 120hz and has HDMI 2.1, to get the full potential out of the Series X.

I have same issue but with my Acer Nitro VG271 144Hz. Xbox series X won’t run at 120Hz 1080p. That must be problem with xbox series x because works perfect with Asus RX480 8GB (144Hz 1080p HDMI2.0). I am very dissapointed. 60hz is a joke.

my sons 1080p msi 27" monitor supports freesync and 144hz over hdmi, it also supports 120hz for ps5 and xbox series consoles too. this has hdmi 1.4. Also tested.

You've stated twice in your post about a PC. What exactly is it you're looking for? A PC or a Monitor?

If it's just a monitor you're after, don't make the mistake of buying 'any monitor' that fits your budget and hope that it works, because as SwordofWhedon has already stated, many monitors have compatibility issues with consoles. Just because a display says it does 144Hz, doesn't mean you'll get 120Hz on the Xbox. Many monitors, especially older models, have their own custom timings that don't conform to the VESA standard that the Xbox expects.

Series S is largely a 1080P console with some games running at 1440P. If you connect a Series S to a 4K display it won't look great. Series X looks good on both 1080P and 4K displays.

https://www.rtings.com/monitor/reviews/best/xbox-series-x#recommendation_264254

series x does not support 144hz, only 120hz.