If the Torah states that a kid can not be cooked in its mother's milk, then wouldn't the prohibition between meat and milk depend on the source of each? That is, the prohibition should be from mixing beef with cow's milk and goat/lamb with goat/lamb's milk.

I can see the extension of "mother" to include all animals of that group, but there is a difference between cow and goat milk.

Answer:

At first glance the verse is simply telling us not to eat a young animal in its mother’s milk.

Based on that meaning your question is completely valid.

However, there are multiple reasons why this verse cannot, and is not, understood on the surface level. This is one of the most cryptic and complex verses in the Torah.

It turns out that this one verse is designed to teach us a myriad of laws, and in quintessential Torah fashion, it teaches us a maximum of laws with a minimum of words.

In its true meaning this prohibition really has nothing to do with a baby and its mother. It is simply worded as such to teach that this prohibition only applies to meat that comes from an animal whose mother gives milk.