There's nothing wrong with milk in and of itself, since milk typically doesn't come from (a) dead animals, or (b) swine. However, many dairy products use haram (or at least questionable) ingredients in their manufacture.
The biggest example would be cheese. The typical cheese-making process requires an enzyme to curdle the milk, and this enzyme is often of animal origin. Rennet and Pepsin, for example, are often sourced from the stomach lining of mammals (including swine). There are non-animal sources of rennet, and there may be some dispute as to the permissibility of non-porcine (e.g. bovine or ovine) enzymes, but as far as I know there's no doubt that when the enzyme is porcine it is haram.
Another example would be yogurt, which can often contain gelatin as an emulsifier. Many yogurts (greek-style yogurts in particular) do not, and gelatin can be halal if it's properly sourced, but given that most gelatin on marketed goods is just labelled "gelatin" there's no way to distinguish that which is derived from pork from that which is halal.
Some thickened cream products can also contain gelatin, including sour cream and whipping cream. Gelatin can also often be found in cheesecake, but I don't think that counts as a dairy product.
I've also heard that the vitamin D3 which is commonly used to fortify milk can come from pork, but I've seen little to no evidence suggesting that this is actually likely (as compared to sheep's wool which is fundamentally halal).
So yes, technically, it's the same situation that with milk being fundamentally halal (assuming, of course, that you're not dealing with pig milk) the only thing you need to worry about is something haram being added to it. But given how common many of these potentially haram additives are in the dairy industry, and how difficult it can be to distinguish halal and haram sources of the same additive, the halal certification makes sense here.
In the particular case of Fleurieu Milk and Yoghurt (the company represented by OP's link), their marketing manager Nick Hutchinson specifically explains:
…our yoghurt doesn't contain gelatine so we can definitely argue the fact that it doesn't need to be certified for Emirates [but] they play it safe. Anything that goes on their planes needs to be certified, so if they're asked, they can automatically answer 'yes'.
In other words, the certification doesn't affect their product in any way shape or form, and isn't even strictly necessary, it's simply a convenience issue for Emirates Airlines (with whom they
have had a major supply contract): Emirates (or any consumer, really) doesn't need to know (or care) about the details of the actual product, they simply need to know that it's halal.