note: I am not a native speaker, but I have been learning Dutch for quite some time.

In this context enig is an adjective meaning snoezig. They're saying that the bird is really adorable.

They're using beest to refer to a huisdier (pet). When addressing a dog, "Braaf beest" would be like, "Good boy" or "Good girl" or "Good doggy."

And no, I wouldn't replace dat with het. They are talking about that particular bird, so it is definitely dat beest, not het beest.

Dat beest, het is echt enig.

I wouldn't try translating it literally into English. Since we know we're talking about a bird (good on you for providing context), "Het is echt enig dat beest" would be something like, "Oh, that is such an adorable bird" in English. (Can't help you with the German.)

By the way, I recall being told somewhere along the way that using enig in this way sounds rather bekakt (posh, snooty).

Het is echt enig. Enig!

https://www.youtube.com/watch?v=wEyqGi_Sd74&t=223s

I don't know how true this is or how much it depends on the individual listener, but I thought it might be worth mentioning.