Ooo! So Kaoru doesn't have a lot of things that they keep but there are a couple things. They used to have a special brush for their tail and ears that was given to them by their mom. When they fled Ul'dah it was lost. After some time in Ishgard Emmanellian ends up having a servant buy them a new fancy fur brush. And they treasure it.
Their most important possession is a pendant in the shape of a Fortemps shield. It was given to them by Count Edmont after Haurchefant's passing to represent the betrothal that didn't get to officially happen. They almost never take it off and it represents the deep love they will always have for Haurchefant.
Eventually Estinien will give them a pendant made to look like his lance as his betrothal to them but it hasn't happened yet.
Some fairly normal ones are two of their outfits. Upon arriving in Ishgard House Fortemps paid for new combat attire for them and they wore that until getting revenge against Thordan. The outfit was too stained in blood after The Vault but they couldn't bring themself to get rid of it either so its still in a box in the Fortemps manor. After that they were given a new outfit by their Ninja master, Oboro and they cherish it as well.