It might be something in the rug that's stimulating the cat, perhaps the fibres or backing. I have two mats that have rubber backings, and my spayed girl rolls on it like there's catnip on it. Fortunately that's all she does...never pees or poops on it. My neuter male has no interest in the mats other than just sitting on them from time to time. I would get rid of the rug. I certainly wouldn't want to reinforce unwanted elimination by leaving it available, in spite of it going with your decor. Eventually you'll have to throw it out anyway, as it's almost impossible to eliminate the odour.
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