It kinda depends.
For instance, Many insurers offer something akin to 'Valuable Property Insurance' (At least that's what mine is called) and for personal use it covers drop/breaks as well as theft.
You typically need proof of ownership; my insurer lets me upload that, so I usually make a point to upload a copy of the invoice/receipt as well as the camera/lens and closeup of the serial number (even better if the invoice has the S/N present!). That's more important for high dollar items typically.
-HOWEVER-
That's for personal use. A while back I actually hit a coverage threshold where my insurer sent me a letter basically saying "Hey, just so you know, you are not covered for business use". (I don't use for business, I just figured it was a cheaper hobby than a boat)
Edited to add:
FWIW the VPP policy is separate from a homeowners policy, however insurers may or may not (depending on state law etc) be able to use a claim on a different policy to impact rates/etc.