It could have been a cat or a dog or anything living. The thing is the idea behind that thought experiment matters which I am explaining below in little brief.
According to the Copenhagen interpretation of quantum mechanics as interpreted by Neils Bohr and Werner Heisenberg, any quantum object is under a superposition of quantum states at any point of time.
For example, we consider a decaying nucleus. One single nucleus in our case. Now, it might decay in the next moment or it may live (not decaying) for many moments to come. So, unless we find it out, the nucleus is under a superposition of quantum states corresponding to both 'nucleus has decayed' and 'it has not decayed'. Quantum state here corresponds to the possible States in which in the system (in our case, the radioactive nucleus) can be. Either it decays or it doesn't in a particular time and thus it is under a superposition of the two quantum states corresponding to both the possible states.
Now as soon as we find out by looking at it (just assume that we can see it) at a particular time, we find out that it has either decayed or it hasn't. In such a case, we can be pretty certain about whether it has decayed or not and that's when the concept of probability of happening of one of the states vanishes and that's what they call is a wave function collapse.
Now, Schrodinger intended to show that such a concept is an absolute nonsense in the real world though it may have a significant interpretation in quantum mechanics.
Considering the cat experiment, we have a cat and a tank of poisonous gas in a box with a radioactive nuclei and a detector of radioactivity. Whenever the nucleus decays, the detector detects it and releases the poisonous gas and the cat dies.
Now as long as the box is closed, a person from outside has no means of finding out whether the nucleus decayed or not or equivalently the cat died or not. Here, the state of the nucleus corresponds to two quantum states 'decayed' or 'not decayed' and their superposition. The decayed state corresponds to 'cat is dead' and 'not decayed' state corresponds to 'cat lives'.
So, as long as a person doesn't open the box, he is not able to find out whether or not the cat is dead (or equivalently the nucleus decayed or not) so, from the perspective of the person outside the box, the cat is in a quantum superposition of states 'living' and 'dead' at the same time.
When he opens the box, he finds out whether it is dead or alive and thus, he is certain about it and thus, there is a wave function collapse and we can be 100% certain whether the cat is dead or alive. That doesn't however tell us at which moment it died (if it at all dies).
So, such a thing is a common experience in the real world and the concept of superposition when applied to the real world yields non sense results though it may interpret quantum mechanics in its own way.
That's what Schrodinger wanted to illustrate with his experiment. It was a thought experiment. No cats were harmed. And it could have been a dog in place of a cat without the ideas behind it changing.