Mm...the LCD X and the Katana offer far different experience. There are far more irreconcilable differences than similarities so comparison between the two will not be of much use. I use my LCD X whenever I want the Audeze sound and the planar magnetic sound and enjoyment in a quiet and comfortable environment. I use the Katana whenever I want a balanced- sounding in-ear in travel mode (outside home). The Katana should be compared with other in-ears, not headphones. But setting aside those differences, are the sound signature similar? Somewhat yes. The LCD X is a neutral-sounding headphones (more neutral than other Audezes like the LCD 3F). Same with the Katana. By neutral, I mean a balanced performance where the highs, mids and lows work in a coherent fashion. Tonally, the Katana is just right.
A lot depends on your needs. If you do not usually have that much time to listen outside your home, you will be better off (and better served) using a more reasonably priced in-ear rather than spending upwards of 1700 usd for a high-quality in-ears. But if you need just one in-ear, the Katana easily earns my recommendation. There are others to consider as well like the Campfire Vega or the HUM Pristine or the AKT8ie, and, if you prefer to go CIEM, there's the Empire Ears Zeus XIV.
I love both in-ears and headphones in equal fashion and regard them as complementary rather than competing product categories.