Let's say I've got a single object with multiple breaks in it (each under 1 ft). Can I cast Mending on each break, or does Mending only work if it can fix a single object with a single casting?
You can fix it by casting Mending multiple times.
This spell repairs a single break or tear in an object you touch, such as a broken chain link, two halves of a broken key, a torn cloak, or a leaking wineskin. As long as the break or tear is no larger than 1 foot in any dimension, you mend it, leaving no trace of the former damage.
If you look at the wording of the spell it's not about repairing the item but the break or tear itself. So you can cast it to repair a tear on the side of your robe, and then again to fix the hems.
Also note the repairs are only physical so it can't restore any magical properties:
This spell can physically repair a magic item or construct, but the spell can't restore magic to such an object.