As I was developing a level 4 character for a 5th edition D&D campaign I was playing with my friends, I settled on a multi-class character using the Druid and Wizard classes. When I went to choose a focus for the character, I saw the "yew wand" in the druid focus list and the "wand" in the wizard focus list. Could the yew wand then be used as a focus for both classes?

You might say something like: "Oh, well, a wizard can use a wand to cast spells, and a druid's yew wand is a wand, so the wizard can use that."

But a wizard's arcane focus isn't just any old wand. A wizard's arcane focus is an item which was specifically created to be an arcane focus. If the druid's yew wand wasn't specifically created to be an arcane focus, we should expect it not to work for a wizard.

Here's the full rules text:

Arcane Focus. An arcane focus is a special item— an orb, a crystal, a rod, a specially constructed staff, a wand-­‐‑like length of wood, or some similar item— designed to channel the power of arcane spells. A sorcerer, warlock, or wizard can use such an item as a spellcasting focus.

Druidic Focus. A druidic focus might be a sprig of mistletoe or holly, a wand or scepter made of yew or another special wood, a staff drawn whole out of a living tree, or a totem object incorporating feathers, fur, bones, and teeth from sacred animals. A druid can use such an object as a spellcasting focus.

So: a super-literal reading of the rules would tell us that a wizard probably can't use a "druid's yew wand" as a focus, because the "druid's yew wand" wasn't "designed to channel the power of arcane spells".

But, if a wizard gets a piece of yew wood and runs it through the process of creating a wizard wand, the result will be a "wand made of yew wood" and a druid can use that just fine.

This is still a bit ambiguous. Xirema notes that the rules for multiclassing tell you that you can only use a focus for spells of the appropriate class, which seems to imply that the designers might have intended for foci to be separate.

Another relevant consideration is that multiclassing tends to result in worse classes than single characters anyway. If we were making this ruling based on game balance, we should probably try to be lenient to the multiclass character, since they're going to have a harder time compared to single classes.

But the gripping hand is that we on stackexchange can still be overruled by your DM. If any rules are ever unclear or ambiguous (and these rules are), the way to resolve them is to ask your DM to make a ruling.

Go ask your DM to make a ruling. : )

Yes.

Because a yew wand is a wand, a Yew Wand would be valid for both Druids and Wizards, so your character could use it for both their Wizard and their Druid spells.

Rules as Written, no.

As written, an arcane focus is an arcane focus, and a druidic focus is a druidic focus. Everything else is just suggestions for descriptive text of what a specific focus might look like.

Rules as Fun, sure!

It totally makes sense for a cross-class character to double up their foci in a way that meets both requirements, so sure, go for it! (Pending DM permission, of course.) The only real, serious difference would be a matter of a few gold -- which is really no big deal for an adventurer of even 2nd level -- and potentially a question of how many hands you're using at any given time.

I don't think the 'number of hands' issue is worth worrying about unless the DM really likes to get fiddly about things like that. Consider the following complaint:

"Wait a minute, you're holding a torch in one hand and your arcane wand in the other, and you used your free object-interaction to put your wand away, so if you want to pull out your druidic focus you're going to have to use an action!"

If your DM might say something like that, then you may have an issue with 'doubling up' like that, as it would remove something your DM is treating as a limiting factor on your spellcasting. If your DM wouldn't say that, then sure, there's very likely no problem with a dual-use wand.

You need two wands; the same wand cannot serve both classes

From the Multiclassing rules, as written in the Player's Handbook, page 163:

Spells Known and Prepared. You determine what spells you know and can prepare for each class individually...

[...]

... Similarly, a spellcasting focus, such as a holy symbol, can be used only for the spells from the class associated with that focus.

While a Yew Wand is almost certainly valid as a spellcasting focus for either a Druid or a Wizard, it's not valid for both at the same time. You'd need to acquire a second wand at the time you multiclass.

If focuses were permitted to be shared between multiple classes, the wording would read something like "can be used only for the spells from the classes associated with that focus"