Is there a way to make the eevee scene look like the cycles one?
The problem here is that you are using Screen Space Reflections.
Eevee uses various shortcuts in order to get good visual renders with minimum render time. In the case of reflections, Screen Space Reflections usually produces results which are good enough in most situations - where everything that is 'Visible' (screen space) is mapped through any reflective surfaces to provide those reflections. This means that only things that are in 'line of sight' need to be rendered but has the downside that if anything would only be visible as a reflection that it will not be rendered.
In your example, the dark shadows you can see in the reflection are those bits of the scene which are not directly visible to the camera - ie, it is behind the cube. Since the reflection provides a view of that 'hidden' section, Eevee cannot render it - and so it shows as an additional dark shadow in the reflection.
The solution is to use a Light Probe (in this case you can use a Light Probe Plane on the floor) to give Eevee a hint as to what to consider for reflections.