I believe they are working to change this limit in the near future, most likely to 250 instead of 100 variants (options are not like variants, the current limit is 3 options).
So for your use case, variants might not be the way to go, but more likely custom attributes (aka product properties). Those are not linked to a SKU, so they can’t affect prices or inventory level (which means that you have one “mother SKU” for all “custom variations”). Custom attributes can be read in the order details (similary as an order note, but assigned to a product instead). It’s also possible to combine both product properties and product variants, it will most likely depend of the way you handle pricing and inventories.
Bar that, custom attributes are more flexible since we don’t have any limit. If you want to create those custom variations and attach an image to each of them, you’d need to hire a developer since it’s likely that there are some specificites to take into consideration.
EDIT : or as @Ivy_4 rightly suggested, you could combine multiple products. In your case, this could be 12 products (for each month) with 28 to 31 variants each (for each day). Here again you’d likely need some type of code in order to add a custom selector for the 12 products (by default you’ll only have a variant selector for the days). It’s possible to make a custom selector that looks like a variant selector, but actually redirecting to the selected product). In both case, I doubt you can achieve a decent result without going through the code, it’s just how it is. A popular example of this is gymshark.com, you’ll notice that their “color variants” are actually redirecting to a new product url.