The question is: does the Natsume Yuujinchou manga address the direct interaction between Taki and Madara?
Per their interaction, the natusme fandom wiki, under Tooru Taki says:
Taki first meets Nyanko-Sensei after he trips her. Nyanko-Sensei/Madara helps Taki get free of the curse, and their bond goes as close to "helpers" or "friends." Because she is overwhelmed by the cuteness of his cat form, she can't resist cuddling Nyanko-Sensei every time she sees him. Nyanko-Sensei tries but often fails to avoid this."
The same resource for Madara's page is empty regarding their character interactions.
The anime version of Natsume Yuujinchou (Natsume's book of friends) is at best vague about whether Taki can understand Madara when he is in maneki-neko "lucky / beckoning cat" form, She certainly thinks he's cute and is always trying to cuddle him. Later in the (anime) series there are instances where it seems they can actually communicate and she knows he is not a "normal" cat - as she also shares in Natsume's secret. On my current "rewatch" of the series (which began because the summer 2022 anime season was mind numbingly uninteresting with the exception of a few standout productions) I've only reached up to season San thus far, so I haven't noted the specific episodes with specific instances of their interaction (if I reach a specific episode I will update this question). But even my first time through the series, I remember having this same question which continued to linger in the back of my mind.
Taki should only be able to see Ayakashi (youkai) when they step within one of her familial Onmyoji circles as is addressed in Zoku episode 6, The Maiden's Circle, but they are vague regarding her ability to interact with them - as she is clearly still cursed even though she can not see the stitch face youkai - and Madara is technically "possessing" the lucky cat statue which is why people can see him.
The question, again, is: does the manga address the direct interaction between Taki and Madara? If you can cite/link something specific which I can google or quote a comment from the author that would be ideal, because I think it's pretty clear the anime material doesn't directly address this and other than "who is Natsume's grandfather?" which I believe was an intentional question mark, there are not a lot of lingering plot holes left open at the series close, so it seems the author was fairly thorough.
Thanks in advance for your informed contributions!