No.
Under the rules for the spellcasting class feature of multiclass characters, this is explicitly covered (PHB, p. 164; emphases mine):
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.
But note that a multiclass spellcaster can use a component pouch to cast spells from all their classes:
A character can use a component pouch or a spellcasting focus ... in place of the [material] components specified for a spell.
(PHB p. 203)
Yes, it can
You can make any non-magical trinket that can fit in your hand, and it disappears after the end of your next turn. One of the trinkets you can make, as listed under PHB 160, is "a whistle made from gold-colored wood" as well as "a tiny silver bell without a clapper"; these are both things that you could make such that they can fit in your hand.
Furthermore, they fit the criteria of "musical instruments," because you can make music with these. While the PHB does list some of the things it considers as instruments, it is not meant to be an exhaustive list:
Tools, PHB 154
Musical Instrument. Several of the most common types of musical instruments are shown on the table as examples.
This means prestidigitation can create a musical instrument that can fit in your hand, and you can use any musical instrument as a spellcasting focus.
Of course, it is inefficient
You must use your action to cast prestidigitation, and then you cannot use your action again to cast another spell. On your next turn, you could use the instrument to cast a spell, but then it disappears at the end of that turn, and you have to recreate it on your third turn. This allows you one spell every other round that isn't prestidigitation (unless it's a bonus action or reaction spell).
Best Answer
This is a complicated issue. To start with, you don't need to use an instrument to cast spells - necessarily.
So, what does it mean to use a spellcasting focus?
So you can use an instrument instead of the actual material components when casting any spell that requires material components. However:
So no, you can't use an instrument to cast spells while holding two weapons - you have to have a hand free. The really important point here, though, is that not every spell has material components. Similarly:
So if a spell requires a material component or a somatic component (or both), then you're out of luck, you'll need a free hand. On the other hand, if a spell only has verbal components (which is the case for many spells on the Bard spell list) then you can cast it while holding a weapon in each hand.
Finally, it's definitely worth taking a look at the War Caster feat, which among other things, grants you this benefit:
This would allow you to cast spells with somatic components while holding your weapons, although you'd still be out of luck for spells with material components.