I’m recreating a character someone made as an NPC and has become one of my PCs.
I don’t see Arcanist as a class option, but reading about it, I made him a multiclass Sorcerer/Wizard. I don’t see Water Breathing as an option in either spell list, even though the site shows it as a 3rd level Wizard spell.
What am I doing wrong? 

Here’s the character https://ddb.ac/characters/37826574/I1nEue

Your character doesn't have access to third-level spells yet. Class level and spell level are different, plus because you've got two classes, you've got the multiclassing rules to deal with as well.

From the optional multiclassing rules:

You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class. If you are a ranger 4/wizard 3, for example, you know three 1st-level ranger spells based on your levels in the ranger class. As 3rd-level wizard, you know three wizard cantrips, and your spellbook contains ten wizard spells, two of which (the two you gained when you reached 3rd level as a wizard) can be 2nd-level spells. If your Intelligence is 16, you can prepare six wizard spells from your spellbook.

Gotcha! Thanks for the pointers!

So, with skills, you can override the rules and give a character proficiency in more than their class and race and background would allow.
Is there a way to do that with spells?
This was an NPC I'm assuming that was created for a purpose, and they want that Water Breathing in there.

You could create a feat that grants the spell, and give the character the feat.

Thanks. I've been reading other threads like this, and it sounds like the only issue is I have to manually track spell slot usage that way.

On another note, I've heard of the Arcanist class before this, and have access to a bunch of the Compendiums, but I don't see it anywhere on the DNDBeyond site. Curious.

Arcanist hasn't been a class since 3.5, which is why it's not represented here on DDB.

Aha! Good to know. 
I haven’t played in a long while.