I teach classification in my job (teach educational groups visiting a relevant institution so cover all ages from nursery to uni groups).
I've done this job for almost 20 years and have played around with both curriculum requirements but also the best way to introduce and expand on these things across the age ranges (allowing for the fact that I only see the kids as a one-off, so while I teach year 1 and year 6 classification it's not the same kids progressing through, if that makes sense, so there is some variation in background understanding).
Vertebrate classification is Year 1.
I do it on the basis of visible characteristics, asking 'What is this animal covered in?'
We do it through first hand experience - touch the fur, describe it, compare it to the snake skin etc. As a classification system it's not perfect but it's a start. If the class asks about animals that don't seem to fit, we can explore that, or if their understanding is good enough. But we also cover why we do this - it's as important to get a handle on why classification is important as it is on how to do it.
Bit later on and we do invertebrates vs vertebrates - and look inside a tortoise shell to see the spine, look at a bat x-Ray, look at and handle mini beasts. And then relate it back to what they already know - how does this fit with the bird reptile mammal thing you already know? Can you remember how to do that bit? Let's refresh and go a bit further with how we group those animals.
The trouble is, any statement used to define a group of animals can almost always be undermined with exceptions. Almost any statement in science in fact. So how do you teach it? This is important and these are the rules but they don't work? So you start with broad statements and ideas and as kids get their heads around it you develop the ideas further. Mammals have fur. Fact. Let's let everybody think about that, experience it, take it in, it's broadly true a d they can broadly speaking see it for themselves so apply it in every day life (that dog has fur, it's a mammal). Now, let's look at a dolphin, no fur - how do we explain this being a mammal? By the end you can be saying 'mammals tend to...' Rather than 'all mammals have.... ' But it's a gradual process.
It's easy as an adult who did science at a high level to see your way through apparent contradictions but lots of kids struggle and then switch off if you start with the the 'I'm telling you how complicated this is' approach right from the start.
Does that make sense?