I was told somewhere that a way to increase spell save DC was by multiclassing and taking a spell in each class multiclassed.

If this is the case, a Sorcerer and Warlock multiclass would be able to add Charisma modifier twice to spell save DC if both have the spell. I'm unsure if it was the ability score modifier or the proficiency that was doubled - are either correct?

No, they do not stack

Per the basic rules on multiclassing, specifically under spellcasting:

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.

Spells you learn through a class are associated with that class, even if they are learnable/castable by other classes. If I learned Hold Monster as a Warlock for example, I would have to cast it as a Warlock spell, even if I multiclassed into Sorcerer or Wizard.

Learning through another class does allow you to choose which to use, but you still only cast from a single class, so you wouldn't be able to add your proficiency/ability modifier twice through multiclassing.

Learning a spell as multiple classes does not empower those spells' Save DCs

You use the Save DC associated with the class you're casting from; you do not gain the ability to add both to your spellcasting.

This is most obvious if you go with a more classical (if less mechanically synergistic) example: the Wizard who sold their soul for Great Arcane Power™.

If a Wizard learns Burning Hands as both a Wizard and as a Warlock (through the Fiend Patron), they'd gain the ability to choose to use either their Wizard Save DC (8 + Prof. + INT) or their Warlock Save DC (8 + Prof. + CHA), but they'd have to choose one of them, not both. This makes sense: as a Wizard, they know the spell through years of training and research, and have perfectly inscribed the formulas necessary to cast the spell as part of their capabilities as a Wizard. Conversely, as a Warlock, they've learned the spell because their connection to their Patron allows them to share the arcane power of their Patron; although both methods produce a discrete spell effect, the methods by which they produce these effects are very different.

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 he used only for the spells from the class associated with that focus.

—Multiclassing, Player's Handbook, pg. 164

The Save DC for each class's spells are specified as a simple formula: 8 + [Proficiency Modifier] + [Spellcasting Ability Modifier]. You would be required to choose which Spellcasting Modifier you use based on which class you're casting from, but there's no specification that you'd add those DCs together.

Additionally, there are no rules that stipulate that it should be possible to add together these modifiers or DCs when casting as a multiclassed character, so therefore, there is no way to do so within the rules.