You're not reading the entirety of the rules. Here's the part you're missing:
See? When that multiclass Wizard gains a Wizard level and adds a new spell, that new spell is limited to a spell that he could know and prepare based on his Wizard levels.
You're focused solely on Slots, which is an entirely different thing than Spells Known and Prepared. But even then, the rules clearly prevent what that Wizard player tried to do:
Note the callback to Spells Known and Prepared, which is the paragraph directly preceding on p 164.
The "Spells Known and Prepared" / "Spell Slots" difference is one of 5e's innovations and it's not totally intuitive, especially to players of previous editions. But trust me, the text is clear. If you're a low-level multiclass Wizard, you'll only be casting low-level Wizard spells*.
* Although, you'll be able to cast those spells in higher-level slots.
Finally, here is the deal-sealer, from the Wizard class description on p 114:
Where in that paragraph is the "Multiclassing Spellcaster: Spell Slots Per Level" table referenced?
No. When a Wizard, multiclassed or singleclassed, gains a level he can add two wizard spells to his spellbook. Each of which must be of a level for which he has spell slots, as shown on the Wizard table.
