Not sure if this has already been discussed, but: this clarification to Magic Initiate is kind of a big deal, if it turns out to work as described above.
Magic Initiate says "You learn that spell and can cast it at its lowest level."
The Errata clarifies that the restriction "Once you cast it, you must finish a long rest before you can cast it again" is limited "only to the casting granted by the feat."
So how does this all interact with spellcasting? It's worth pointing out that the feat does NOT grant a Spell Slot, 1st level or otherwise.
The rules for Multiclassed spellcasters are irrelevant, here. Magic Initiate does not grant a level in a spellcasting class.
So, we only have the rules from each spellcasting class's "Spellcasting" description. And the rules in Chapter 10: Spellcasting.
This passage from Chapter 10, "Known and Prepared Spells" is interesting:
The Magic Initiate feat says you "learn" the spell. That sure sounds like "always fixed in mind." Perhaps it should be treated that same way.
The Wizard's "Preparing and Casting Spells" description says "The Wizard table shows how many spell slots you have to cast your spells of 1st level and higher. To cast one of these spells, you must expend a slot o f the spell’s level or higher."
The crux, then, is if the spell you "learn" from Magic Initiate counts as one of "your spells." I'd rule YES. So, you can use a Wizard slot to cast it.
However, that section furthermore says "You prepare the list of wizard spells that are available for you to cast. To do so. choose a number of wizard spells from your spellbook equal to your Intelligence modifier + your wizard level (minimum of one spell)."
Magic Initiate effectively bypasses the normal limit on Prepared Spells or Spells Known. It bypasses one of the main restrictions on spellcasting classes. The feat should probably be renamed "Magical Talent."