Tested in Krita 5.3.3.
0. Import the solid-colored image & the background(s) into Krita. (make sure to scale them so the images fully overlap).
Create a new group layer in the "layers" panel (click the arrow next to the plus), then drag all the images into it (make sure the solid-colored image is at the top of the list).
Make sure to select the image you want to pattern, then, at the top of the "layers" window, where it says "Normal", use the scroll wheel or click it to swap between different blending modes.
Find one that keeps the solid color, but has a background pattern. (see image on the right for example).
Right-click the solid-color image in the layers panel, then click "select opaque" in the menu. Switch to the select tool, then right-click inside the selection and click "invert selection". You then use the paint brush with the eraser tool selected to remove the excess background. (note: when you do this step, it means you can't move the background image, so make sure you're sure about the position, unless you want to spend a ton of time undoing).
Done! You can then explore more blending options, or apply filters to the image to adjust it.
This example logo is the Kanro logo.
This is quite a simple process, though, finding the right image, or the right blending effect may take some time for you. make sure to experiment!
While this effect might work for more than 2 color hues, it's recommended to use only 2. (color + it's shades & black to white).