What would be the balance implications of allowing multiclassing with the same class (e.g. for access to multiple subclasses)?

The rules for Multiclassing on page 163 of the PHB state:

With this rule, you have the option of gaining a level in a new class whenever you advance in level, instead of gaining a level in your current class.

By RAW, it seems clear you cannot start out at level 1 again in your current class in order to achieve something like a Sorcerer (Draconic Bloodline) 1 / Sorcerer (Divine Soul) 1.

Would it be balanced to allow this anyway? Which problems would occur?

Assume that all other rules for multiclassing remain intact, such as the spell slot calculation or the limitation of multiple instances of Extra Attack.

Related: the same question for Pathfinder, clarification that this is not allowed by RAW