An exception to the exception
The crucial context your question is missing is the Identifying a Magic Item section of the DMG (p. 136) which says (emphases mine):
Whatever a magic item's appearance, handling the item is enough to give a character a sense that something is extraordinary about it. Discovering a magic item's properties isn't automatic, however. The identify spell is the fastest way to reveal an item's properties. Alternatively, a character can focus on one magic item during a short rest, while being in physical contact with the item. At the end of the rest, the character learns the item's properties, as well as how to use them. Potions are an exception; a little taste is enough to tell the taster what the potion does.
As this passage explains, normally the properties of magic items are not revealed by simply handling them, but only the fact that they are, indeed, magical. Normally an identify spell is required to quickly discern what a magic item actually does.
However, potions are an explicit exception to this; each type of potion has a characteristic look (described in its DMG entry) so that they can be recognized from previous experience. Even if the specific potion in question has not been encountered before, a little taste of it reveals its properties to even a naïve taster. Thus, an identify spell is explicitly not needed in the case of potions.
Armed with this context, we are now prepared to make sense of the description in the potion of poison. Here, we are told that in fact we do need an identify spell to determine that the potion is poison. Thus, it is an exception to the rule that one doesn't need the identify spell to identify potions (which is itself an exception to the rule that one does need the spell to identify magic items).
Further, the fact that a poison potion is "masked by illusion magic" is precisely because it "looks, smells, and tastes like a potion of healing". Normally a potion is identifiable through one's senses alone, but here the illusion magic is disguising the true taste and appearance so that the taster is misled via the usual means of potion identification.
To me, the following two interpretations seem just as likely:
[1] The "illusion magic" can trick spells such as Detect Poison and Disease, but not Identify.
The illusion magic does not serve to confound other means of identifying the potion, just the 'standard' one of appearance and taste, and referring to the identify spell does not imply that that is the only spell which will work. As a magical means of locating poisons, Detect Poison and Disease will not be fooled by the false 'look, smell, and taste' of the potion, because those are not what it is using as criteria; the spell will correctly identify the potion as a poison.
To me, the following two interpretations seem just as likely:
[2] It was obvious that Detect Poison and Disease would work and only the interaction with Identify needed to be mentioned, since Identify usually doesn’t work on poisons.
It is explicit in the description of Detect Poison and Disease that it will work. But it is not that Identify doesn't usually work on poisons - Identify will work on any (non-cursed) magic item. Rather it is that Identify isn't normally needed on potions, but poison potions are the exception.