There is definitely some pattern to the naming of these ships. The enemy ship classes are named in Iroha order corresponding to the first 18 characters of the Iroha, a classic Japanese poem which uses all 47 classical hiragana* (excluding ん but including ゑ and ゐ) once. This is an old-fashioned way of ordering hiragana (modern Japanese usually uses Gojūon ordering). It can be used to label things in order without resorting to numbers, similar to how one might use letters A, B, C, etc. For example, the series Sayonara Zetsubou Sensei uses the same ordering for classrooms, though in modern day Japan classrooms would usually be labelled by letters. The main difference between this and other ordering systems is that Iroha order feels more authentically Japanese and old-fashioned than the other options.
A table illustrates the relationship between the numbers and the kana. From the wiki
Ship Class Normal # Elite # Flagship # Destroyer I-class 501 514 564 Destroyer Ro-class 502 515 552 Destroyer Ha-class 503 516 553 Destroyer Ni-class 504 514 Light Cruiser Ho-class 505 515 554 ...
You can see that the numbers are ascending, particularly in the leftmost column. I'm not sure the right columns are correct since there's some repetitions, but it would be a lot more work to check everything than it's worth doing here. Anyway, at least the normal numbers very clearly count up from 500, while the class kana progress in the Iroha.
That still doesn't explain how the ships themselves are ordered, though it does say how they're named once you know the ordering. The ordering of types is pretty clear. All the Destroyers come before Light Cruisers, which are before the Torpedo Cruisers, etc. For the most part, it goes from smaller to larger ships. It's not exactly the same as the order here for your available ship types (for instance, battleships come after carriers, rather than before), but it's similar.
How the ships are ordered within each type is a bit less clear. I think that most enemy vessel classes are based on a class of ship available to you. For example, Battleship Ru-class seems to be based on the Nagato-class, while the faster but less armored Battleship Ta-class has more similarities with the Kongou-class battleships, and the Aviation Battleship Re-class is based on the Fusou-class aviation battleships. The order there is compatible with the order in the Ship List, in that Nagato-class comes before Kongou-class, which comes before Fusou-class (at least prior to any remodeling).
I suspect for the other types of ships, there's a similar relationship, but I don't actually know which enemy class corresponds to which of your classes, and some don't seem to match well at all. I'm not even sure the creators really put that much thought into it. If there is a nice correspondence, I suspect someone has compiled this somewhere before, if possibly only in Japanese. However, I didn't immediately find it while searching, and it's a tangential enough that I'm not sure it needs to be stated here. Even if there isn't an exact correspondence, it's clear that the enemy ships are numbered in a way that at least makes sense when you compare it to your ship list.
Anyway, at the end of the day, there's no real meaning to the individual names. In terms of literal interpretation, they could have just as well used "A-class", "B-class", etc. (nevermind that those have a different meaning in English). The use of Iroha order here is consistent with the general mood of KanColle as a semi-historical game. It probably isn't worth reading much more into it than that.
Note that the ship classes are actually written in Katakana, e.g. 駆逐イ級 rather than 駆逐い級. This doesn't really change anything significant, and one can just as well write the Iroha in Hiragana or Katakana. In this context, it's similar to writing in italics in English. There's a bit more to it than that, but the subtleties between Hiragana and Katakana use in this sort of context aren't really worth going into here.