Can a multiclass sorcerer/cleric use the Metamagic class ability with non-sorcerer spells? For example, using Twinned Spell when casting cure wounds to heal two creatures instead of just one?
Sorcery points are not restricted in which spells they can affect.
Other class features similarly work between classes, as the classes merely grant abilities and few of them state that they have any form of exclusivity. Metamagic abilities and other methods of spending Sorcery Points are not restricted to only affecting Sorcerer spells.
Considering that in addition, Spell Slots are universal, it doesn't appear to be game-breaking since multi-classing also limits the ability to get more sorcery points.
Our game group had the same discussion and our GM found this and that convinced him ("Multi-classing", PHB, p. 163):
Multiclassing allows you to gain levels in multiple classes. Doing so lets you mix the abilities of those classes to realize a character concept that might not be reflected in one of the standard class options.
With the statement it sounds like it is in the spirit of the rules to allow sorcery points to be used on non-sorcery spells.
Yes, sorcerers can use Metamagic on non-sorcerer spells
None of the descriptions of the options for the sorcerer's Metamagic feature state that they are limited to sorcerer spells (i.e. spells you know/cast as a sorcerer); they can all simply be applied "when you cast a spell" (or, for Empowered Spell, "when you roll damage for a spell"). None of them are restricted to sorcerer spells specifically - and nothing in the multiclassed spellcasting rules suggests that Metamagic doesn't work for multiclassed characters' non-sorcerer spells.
By the same logic, Metamagic options can also be used on spells cast using racial traits, feats, magic items, or any other source.
Rules designer Jeremy Crawford unambiguously confirmed this interpretation in an unofficial tweet from January 2015 responding to a specific version of this exact question (even using the same multiclass example that you used in the body of your post):
Can my multiclass cleric/sorcerer use metamagic effects on cleric spells?
Crawford's answer reiterates what's already clear based on the rules; since the Metamagic feature doesn't limit itself to sorcerer spells, you can use Metamagic options on any spell you cast.
The same logic applies to the paladin's Divine Smite feature, which lets you expend a spell slot to deal additional radiant damage on a melee weapon attack with a weapon:
If I multiclass into Paladin, can I use up any class's spell slots for Divine Smite?