Warlocks and Sorcerers share Charisma as their spellcasting ability and both can use an arcane focus. The Rod of the Pact Keeper is a focus for warlocks. Can a multiclass Warlock/Sorcerer use the Rod of the Pact Keeper as a focus for both spell casting classes but without the extra abilities the rod give the warlock?

Yes.

Let's make a little checklist:

• 
  Requires Attunement by a Warlock. You are a warlock, check.


• 
  It's an arcane focus. Both of your classes can use an arcane focus, check.

There's no reason you wouldn't be able to use as an arcane focus for both classes since you have the only two requirements of the item. As you said, your sorcerer spells can't benefit from the bonus but the special ability only requires you to hold the rod, use an action and have a spent warlock spell slot.

Rod of the pact keeper isn't a spellcasting focus.

Here's the complete, current rules text for the rod of the pact keeper:

While holding this rod, you gain a bonus to spell attack rolls and to the saving throw DCs of your warlock spells. The bonus is determined by the rod’s rarity.

In addition, you can regain 1 warlock spell slot as an action while holding the rod. You can’t use this property again until you finish a long rest.

Nowhere does it say that the rod can be used as a spellcasting focus for any class.