I live somewhere where, a year ago, sadly, an enormous complex of fires ravaged neighborhoods over 3 counties. Although the flames didn't quite reach my town, I evacuated with my 6 cats. I then closely followed the work of the multiple groups of people who spent months rescuing and trapping cats in and near the burned areas (some even ongoing).
These folks came up with methods of cataloging the cats they found. There were many hundreds of cats. A good number were from pre-existing feral colonies or were strays that had been abandoned before the fires. But a lot were animals separated from their humans because of the fires.
Another issue is that most of these animals were filthy (the ash in the air and covering every outdoor (and indoor) surface for miles was insane, and hard to wash off) and some had been injured in ways that made it difficult to recognize them (often because of bandages in medical facilities). But, even for cats that more or less looked the same as they did pre-fires, identification was difficult.
The rescuers took pictures of each cat they found and had records indicating the medical or shelter facility the cat was taken to. If the cat was found by a regular person, the facility would create the records. (This system was not perfect and many people would just care for cats themselves.)
The primary way they classified the cats was by color. While many cats were put into more than one color category, a more detailed system, like the one your story would use, would account for the variations and have more choices. So instead of "tabby," you might have "gray tabby" and "gold tabby." And you'd have groupings for cats of more than one color.
You might think the classification of fur length (short, medium, long) would be primary, but I think the physical issues resulting from the fires made that impractical. Of course fur length was noted in descriptions. Along with gender, intact status, size, age, and anything else of note.
And this brings us to markings. A lot of cats (none of mine, but most of the fire cats) did have markings of some sort. Many desperate people searched for their pets using online resources and info on found pets got posted to several websites and Facebook groups.
Using the pattern of the markings became an art form. There was no good nomenclature for it, but people worked endlessly to match pictures of lost cats with pictures of found ones (dogs were easier to ID but a lot of the lost animals were livestock, where there were similar issues to cats).
I saw so many heart wrenching cases where a distraught person would post a picture of her/his cat and a dozen people would say "I just saw a picture of this cat in the found section!" but it would be the same, wrong, cat the person had seen a dozen times. Sometimes the pictures of these unrelated cats would be so close that only the owner could tell the difference, and only in person (which they did, over and over again).
In one case that made the news, a shelter prematurely adopted out a fire cat and the new family refused to give back the cat when the real family came forward (losing your house is traumatic and it took them a few weeks to track down the cat). The markings from photographs matched the cat but it wasn't enough to prove identity. Authorities finally used DNA. It matched and the adoptive family was compelled to return the cat to its family.
What's the takeaway? Color and markings can change over time. With age, with cleanliness, and with injury. The only reliable method for matching a cat with its human family turned out to be microchipping (and that failed much of the time because people move and don't update their contact info...a lot of the found cats had their names in their listings, but no way to find their families).
I've now chipped all my cats and it's the only method I could see for a high tech society to identify cats who walk by a camera. Even for ID done when booking a crime suspect, for example, I don't think markings will work in many cases, because they change over time. But that will depend a lot on the way your cats are marked...if everyone has prominent markings, clear enough to be spots and stripes and etc, then maybe. If it's like real-life cats, it won't be good enough.