Under the rules for the spellcasting class feature of multiclass characters, this is explicitly covered (PHB, p. 164; emphases mine):
Each spell you know and prepare is associated with one of your classes, and you use the spellcasting ability of that class when you cast the spell. Similarly, a spellcasting focus, such as a holy symbol, can be used only for the spells from the class associated with that focus.
But note that a multiclass spellcaster can use a component pouch to cast spells from all their classes:
A character can use a component pouch or a spellcasting focus ... in place of the [material] components specified for a spell.
(PHB p. 203)