How and where should I display the category color?

It's not easy to answer this question because it's unclear what level of subtlety is intended to be achieved; only the person behind the project knows, with full knowledge of the functionality, users, etc.

This answer is only a formal analysis of the components at a graphic level.

The main problem posed is where to place a second color that refers to a functionality. What I would propose is to strip the design of all the elements that influence the decision and try to visualize its formal basis.

In essence, it's a background with holes or windows or a background with figures. See the law of figure/ground.

The similarity in color between both components, while it makes perception difficult, the essence remains the same: figure/ground or ground/figure (left graphic). There is a third element that attempts to clarify doubts, although it creates a double interpretation: light. If we interpret it as falling from the left, we will see figures on a plane. If it hits from the right, a plane with holes (right graphic):

From here on, any added element can benefit or harm this perception. For example, what is the incidence of a second light or what is the direction of the light? If we add a color to each position of the light (Second Theme, Style 2) to this change, we are clearly harming a clear and immediate perception.

With this clear as a basis, we can look for solutions.

For example, use any element incorporated not arbitrarily but to favor the general perception:

I really like the reasoning of the previous answer by Danielillo. I think what you are refering to is a problem of grouping, which is what the Gestalt principles were invented for. You are already using the "Principle of Proximity" by arraning the servers of one category next to each other. And you are attempting to use the "Principle of Similarity" by using equal colors. But I think by using the visual channel of color twice in the design - once for the category, once for the server status. This might be confusing / make it less useful for the grouping. Maybe there is another way you could make the servers of one category similar like a common icon ?

I think especially the "Principle of Enclosure" could be leveraged for your design, by enclosing all servers of one category e.g. by adding a third level of depth: background, category card, server card with the category card spanning all the area also behind the server cards. If you want to add color here, maybe the category card can have a light blue background color, also to distinguish status colors as lines with category colors as areas maybe.

Another option if you don't want too much of a change and want to stay with the idea of using colors for the categories you might color the background area and icon of each server which would also at least make a difference between status = color of a line, category = color of an area.

As always you of course know more about the specifics of the projects and can therefore decide if my thoughts are reasonible or not, but I hope they can maybe give you a new perspective :)