In D&D 5e, if for example, I chose a Wild Magic Sorcerer, and later on wanted to become a Wizard, would I get to choose another subclass for Wizard?
Yes, you get a subclass for each class
Gaining a subclass is one of your class features (gained somewhere between 1st and 3rd level, inclusively). When multiclassing, you gain all class features of both classes (with some exceptions around spell slots, Extra Attack, etc.; see the multiclassing rules on proficiencies and class features for details).
So when you take your second level of the wizard class, you gain the Arcane Tradition feature, which lets you choose a wizard subclass.
Keep in mind that you cannot choose multiple subclasses for the same class.