Bards can only use instruments as foci (and subclass specific additions). The can't use wands etc.
"You can use a musical instrument (see chapter 5, "Equipment") as a spellcasting focus for your bard spells."
The instrument serves as an arcane focus, just like the ones that wizards, sorcerers and warlocks use. It doesn't matter if they can use the other kind. They still have an
arcane focus to interact with the
arcane magic category.
Furthermore, druids can't use holy symbols either. So are they not divine casters?
"You can use a druidic focus (see chapter 5, "Equipment") as a spellcasting focus for your druid spells."
They use a different
divine focus than a holy symbol to interact with the
divine magic category.
Further furthermore, everyone can use a component pouch.
Because........................................................................a focus substitutes for components! If you have the component, you don't as a wizard need an
arcane focus to interact with the
arcane magic category. And this isn't relevant, since regardless of the existence of components, there are still mechanical rules involving foci and the categories of magic. Components cannot erase that.
And the restriction on foci is based on class not on source of magic. That's why the Eldritch Knight and Arcane Trickster can't use foci despite borrowing the exact same arcane magic as the wizard.
Regardless of where the restriction is, it still interacts with the specific category of magic that class uses and no other. If the categories did not exist, any class that can use a focus would be able to use any focus to do it. A wizard could use a holy symbol, a warlock a druid focus, and a druid a wand. That they need a specific focus that only interacts with their category of magic is proof beyond doubt that the categories do exist. Mechanical interactions do not lie.