You have set Refraction Depth to zero.
It is not clear to me if values below are correct, but you get rid of the "shadow" thing.
Go to Properties Editor > Material > Settings > Refraction Depth and play with value.
In your case 0,1-0,4 m works. It seems to be distance from glass to Monkey.
It seems to be more complicated ... it depend if it should be glass plate or deep water. In second case probably make sense increase to 4 m, that is distance from surface to bottom.
In that case you have to also increase Thickness under SSR Render Properties.
My theory notes behind
The "weird shadow" (as you call it) is a part of bottom plane surface that is not visible from current point of view on background plane. You can see actually through the plane what is behind in that place.
You can actually confirm that by your self. Check where is the issue located on glass plane and switch to wire mode. Real monkey object is placed exactly on the same place on background plane from current point of view .
I guess by default this is some kind if refraction clipping to keep Eevee real-time.
Similar thing happens when you try set reflected floor with object howering above - in a floor is not reflected bottom part of mesh that is hidden from point of view. Like here – Screen Space Reflections in EEVEE
See also ...
Manual - Refraction Depth