When the Bible divides animals into fish, birds, and land animals (wild and domestic) and then talks about "classes" between them, it is making it obvious that animals have natural classifications since they were created separately and there are bounderies between them.
In Hebrew, other terms that resemble the same concept are also used, like here:
Lev. 19:19a ‘You should keep my statutes: You must not interbreed two sorts of your domestic animals. ...’
This different word is the Hebrew כִּלְאַיִם which is been translated "mixture of various classes" or similar, and it can also be applied to different seeds to be planted in the same field, different fabrics to be combined in the same item of clothing.
So evidently the animals were separated into classifications.
These passages are very interesting and show other classifications of animals that the Israelites needed to seriously consider:
Deut. 14:9 “Of everything that is living in the waters, you may eat these: Anything with fins and scales, you may eat. 10 But you must not eat anything that has no fins and scales. It is unclean for you.
11 “You may eat any clean bird. 12 But you must not eat these: the eagle, the osprey, the black vulture, 13 the red kite, the black kite, every kind of glede, 14 every kind of raven, 15 the ostrich, the owl, the gull, every kind of falcon, 16 the little owl, the long-eared owl, the swan, 17 the pelican, the vulture, the cormorant, 18 the stork, every kind of heron, the hoopoe, and the bat. 19 Every winged swarming creature also is unclean for you. They should not be eaten. 20 Any clean flying creature you may eat.
Lev. 11:1 Then Jehovah said to Moses and Aaron: 2 “Tell the Israelites, ‘These are the living creatures of the earth that you may eat: 3 Every animal that has a split hoof and a cleft in its hooves and that chews the cud may be eaten.
4 “‘But you must not eat these animals that chew the cud or have a split hoof: the camel, which chews the cud but does not have a split hoof. It is unclean for you. 5 Also the rock badger, because it chews the cud but does not have a split hoof. It is unclean for you. 6 Also the hare, because it chews the cud but does not have a split hoof. It is unclean for you. 7 Also the pig, because it has a split hoof and a cleft in the hoof but does not chew the cud. It is unclean for you. 8 You must not eat any of their flesh or touch their dead body. They are unclean for you.
9 “‘This is what you may eat of everything in the waters: Anything in the waters that has fins and scales, whether in the seas or in the rivers, it you may eat. 10 But anything in the seas and in the rivers that has no fins and scales, among all the swarming creatures and of every other living creature that is in the waters, it is a loathsome thing for you. 11 Yes, they should be loathsome to you, and you must not eat any of their flesh and you must loathe their carcasses. 12 Everything in the waters that has no fins and scales is a loathsome thing to you.
13 “‘These are the flying creatures that you are to loathe; they should not be eaten, for they are loathsome: the eagle, the osprey, the black vulture, 14 the red kite and every kind of black kite, 15 every kind of raven, 16 the ostrich, the owl, the gull, every kind of falcon, 17 the little owl, the cormorant, the long-eared owl, 18 the swan, the pelican, the vulture, 19 the stork, every kind of heron, the hoopoe, and the bat. 20 Every winged swarming creature that goes on all fours is something loathsome to you.
21 “‘Of the winged swarming creatures that move on all fours, you may eat only those that have jointed legs above their feet for leaping on the ground. 22 Of these you may eat: various kinds of migratory locusts, other edible locusts, crickets, and grasshoppers. 23 All other winged swarming creatures with four legs are something loathsome to you. 24 By these you would make yourselves unclean. Everyone touching their dead bodies will be unclean until the evening. 25 Anyone carrying any of their dead bodies should wash his garments; he will be unclean until the evening.
26 “‘Any animal that has a split hoof but does not have a cleft and does not chew its cud is unclean to you. Everyone touching them will be unclean. 27 Every living creature that walks on paws among the creatures that walk on all fours is unclean to you. Everyone touching their dead bodies will be unclean until the evening. 28 The one who carries their dead bodies should wash his garments, and he will be unclean until the evening. They are unclean to you.
29 “‘These are the swarming creatures of the earth that are unclean to you: the mole rat, the mouse, every kind of lizard, 30 the gecko, the large lizard, the newt, the sand lizard, and the chameleon. 31 These swarming creatures are unclean to you. Everyone touching their dead bodies will be unclean until the evening.