It isn't correct.

You are right, it's wrong. It is this tts voice issue. On ivona.con website when you choose Maja voice (it's duolingo one) and type in "Kot" it's pronounced wrong. But when you type "Kot a nie kod" it's pronounced ok.

It sounds like / kod/ because that's how you pernicious it

A "t" is a soft "d" in a linguistic sense. People are liable to hear one or the other sometimes.

I have studied a bit of Russian and this is more evident there as the unstressed parts of words tend to get softer version of sounds "z" to "s", "d" to "t" and "v" to "f" among others.

What does it mean in German? Does it refer to a different type of cat? ( ͡° ͜ʖ ͡°)

The female voice says very clearly "kod", which is why the audio exercises are disabled.

The new, male voice says very clearly "kot", though.