I have a product listing page with multiple products and each product image has an overlapping image on it. It would be like this example where the basketball is the product image and the question mark is a separate image on top of it. (jsfiddle)
If the overlapping image does not have ALT tag defined, but the main image has an ALT tag defined will it impact SEO for this particular page?
Simply having overlapping images has no effect on SEO in general. This is the kind of minutia you shouldn't be worrying about — rather make sure that the user experience is correct and it is clear what and why you are doing it.
There's still some things to note: Google sees two separate images, and you should definitely provide alt text for both.
I'd disagree that this would be of no concern. Your biggest concern I'd imagine is going to be your mobile experience. Say one image is linked but another is not (I'm sure neither are but stay with me), in this case a crawler (or user) gets to the page can't click on one image versus the other.
In GSC you may see that you're getting a warning for images clickable elements being too close or something of the sort. It's just a mess and you should be worried about it in my opinion.
If the images need to be one, I'd do so in a photo editor, export the single image and add the alt to that.
One more note, though this isn't going to skyrocket your organic traffic overnight I can't stress enough building your site and processes correctly the first go around. It's a minutia until you get hit with an warning or have a poor mobile experience reflected in your rankings.
