cora's obsession would certainly be less obvious than law's. i think he'd really admire law for the man he's become, which includes bragging non-stop about how smart law is and what an amazing doctor/captain he is.
being so caught up thinking about law would also make cora way more clumsy and accident prone than usual, he just can't get law off of his mind lol
i also really like the idea that, just like law expresses his devotion to cora physically through his tattoos, cora could develop some kind of pride for the scar that law gave him when they first met. it's proof of how much their relationship has improved, but having a physical mark left by law on his skin also makes him feel like he really belongs to him. out of all the wounds that cover his body, that's the only one he treasures and remembers fondly (can you tell that i love the stabbing scene lol)