Obviously not, the Bible indicates different kinds at creation.
The new Testament says that there are four types of flesh.
1 Corinthians 15:39
Not all flesh is the same: People have one kind of flesh, animals have another, birds another and fish another.
Then Genesis names 3 groups within the flesh of animals.
24 Then God said, “Let the earth bring forth the living creature according to its kind: cattle and creeping thing and beast of the earth, each according to its kind”; and it was so.
Is Paul saying there are four types of flesh, or is he simply providing examples of differences between different creatures, because Paul's point is to illustrate that bodily resurrection isn't simply rescuscitating the body, but a transformation of the body. Passing from mortality to immortality, death to life, what is sown perishable is raised imperishable. That the kind of flesh we have now will be raised, but with a different "glory".
"
But someone will ask, “How are the dead raised? With what kind of body do they come?” You foolish person! What you sow does not come to life unless it dies. And what you sow is not the body that is to be, but a bare kernel, perhaps of wheat or of some other grain. But God gives it a body as he has chosen, and to each kind of seed its own body. For not all flesh is the same, but there is one kind for humans, another for animals, another for birds, and another for fish. There are heavenly bodies and earthly bodies, but the glory of the heavenly is of one kind, and the glory of the earthly is of another. There is one glory of the sun, and another glory of the moon, and another glory of the stars; for star differs from star in glory.
So is it with the resurrection of the dead. What is sown is perishable; what is raised is imperishable. It is sown in dishonor; it is raised in glory. It is sown in weakness; it is raised in power. It is sown a soulish body; it is raised a Spiritual body." - 1 Corinthians 15:35-44a
Your quote from Genesis really only emphasizes the point I made. "Kinds" are not biological categories, but a way of saying "this and that", of making a distinction. So Genesis 1:24 mentions different kinds of creatures, not exhaustively, and not as discrete biological categories. Genesis 1:24 does not mean there are only three "kinds", consisting of "cattle", "creeping things", and "beasts of the earth", it simply presents an assortment of known creatures. Because ancient people were quite capable of looking around them and go, "Hey, that's a thing" and "Oh, that's another thing". That's a dog, that's a fish, thus are two different things, two "kinds".
A "kind" could be all cats, for example "cats" would describe the "kind" "cats". But there could also be calico cats, that's a kind of cat. That cat with a spot on its head, that's a kind of cat too. Tigers? Tigers are a kind of cat, a bigger kind of cat.
Bats, for example, are a "kind" of flying creature, an
'owph, a thing with wings. Which is why bats are a kind of "bird" in Leviticus 11:19, because
'owph doesn't mean "avians" it simply means "flying things" or "things with wings". Bats are things with wings, and vultures are things with wings, so they are both the same "kind" of thing; and within that kind--
'owph--there are other kinds--vultures, hawks, pigeons, owls, etc.
This kind of gets me to one of my problems with YEC, YEC doesn't let the Bible actually say what it says. Because in order for YEC to "come out right", the Bible has to be changed to say what it never says, and to mean what it never meant.
The Bible doesn't talk about biology, it talks about practical observations made by ancient people; and those observations are not perfect. It's why they could speak about the "vault" of heaven, or a "firmament"--because it looked as though there was a dome covering the earth. If you walk to a place that's really flat, and see the sky touch the horizon in all directions. Why not imagine that the earth is flat or flattish, and that the sky is a literal dome. That description accurately reflects how someone might observe the world without knowledge otherwise.
And it's okay if the Bible mentions those observations. Because it is an authentic, and earnest expression of God's people making sense of their world. It really doesn't matter that an ancient Hebrew writer imagined the cosmos to look and be one way, and for them to be wrong because they simply didn't know otherwise. That doesn't cause any injury to the divine inspiration and power of Scripture to proclaim the Crucified and Risen Christ, Jesus of Nazareth born of Mary, the Eternal Son and Word of God who became man, being born during the reign of Herod the Great and who suffered and was put to death under Pontius Pilate.
What matters is the Gospel, not the scientific accuracy of the Bible. God's Word to us is Jesus Christ.
-CryptoLutheran