My wife and I filed an I-90 for an incorrect green card several weeks back. My understanding is that we won't get a receipt notice. How will we know if they approve or deny it? Will they update the case the incorrect card was based on? Or will we just have to see if they mail us something eventually?

You are already approved, there is no denial of an I-90, this is to replace a card.

If you need proof of status, you can schedule an info pass appointment, and request an I-551 stamp in passport, so that the passport can act as a temp green-card.

They may "Touch" the case, but I doubt they will indicate anything else online.

You should still have I-751 NOA letter, and old CR-Green-card, but if you are past the I-751 NOA letter's 1 year period, then get the I-551 stamp.