Both Warlock and Sorcerer use an arcane focus for their spell casting. Improved Pact Weapon lets the Pact of the Blade Warlock use a summoned weapon as their spellcasting focus.
As a multiclass Sorcerer Warlock can I use this same summoned weapon as my arcane focus for my Sorcerer class spells or must it be a different arcane focus?
A Sorcerer uses an Arcane Focus by default (PHB p. 101), no mention of weapons. Improved Pact Weapon says:
You can use any weapon you summon with your Pact of the Blade feature as a spellcasting focus for your warlock spells.
Since weapons cannot normally be used as a focus and this feature explicitly states that it applies only to a subset of all spells, it does not apply to other spells.
Just because two things have one property in common does not mean they have other things in common. An example: you can use both a shovel and a broomstick as improvised weapon. That doesn't mean that you can use a broomstick to dig a hole.
You would be able to use it to cast warlock spells but not sorcerer spells.
Pact weapon states:
You can use any weapon you summon with the Pact of the Blade feature as a spellcasting focus for you warlock spells.
If you use an arcane focus as your spellcasting focus it would work for both sorcerer and warlock because they both just happen to use the same type of focus.
Sadly because it specifically states warlock, you can't cast sorcerer spells with it.
Improved Pact weapon (Xanathar's Guide to Everything, p57) explicitly states that it works for Warlock Spells.
You can use any weapon you summon with your Pact of the Blade feature as a spellcasting focus for your warlock spells.
As a result, it cannot normally be used as an arcane focus for your Sorcerer spells, unless the weapon in question (such as a Staff already usable as an Arcane Focus), or a specific feature of the chosen Sorcerer subclass permits it.
