My friend has a multi-classed wizard 5, sorcerer 4. They recently took their 5th level in wizard, and took Steel Wind Strike (5th level) as one of the two spells they can add to their spellbook upon level up.
The multiclassing rules for spellcasting state that you determine the spells that you can prepare and cast for each class individually, however, this does not appear to apply to the new spells that you can add to your spellbook, the only requirement being that you must have a matching level of spell slots.
Spells Known and Prepared. 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.
Learning Spells of 1st Level and Higher Each time you gain a wizard level, you can add two wizard spells of your choice to your spellbook. Each of these spells must be of a level for which you have spell slots, as shown on the Wizard table.
Is my friend's action permissible by RAW, that they can add spells to their spellbook that they won't be able to prepare or cast?
Acknowledging the errata: That specifically deals with copying spells into a spellbook, which is a feature distinctly separate from the one that allows you to gain two spells into your spellbook from leveling up as wizard.