Do physical bitcoins or altcoins actually carry value?
The place you bought from, ShibeMint.com, sells nothing more than keepsake tokens.  They are made of copper or silver and are worth only what someone else is willing to pay for them.  They will not include a private key that allows you to then spend Dogecoin somewhere else. These are simply mementos and not a crypto currency.

Quote ShibeMint.com, these physical dogecoins are for 


  displaying additional shibe pride, starting an exciting conversation, more accurate decision making, extra inspiration, a lovely gift for a dogecoin enthusiast, attracting quality shibemates


There is nothing to stop another website from creating and selling their own version of Dogecoins.  In fact, here's another that does.

You can purchase physical bitcoins or altcoins with or without a private key that would allow you to spend that currency.  Here is a full list of physical crypto coins.  Without the private key, the crypto coin is worth the metal/plastic/material from which it is minted.  

That said, it will still carry the value that someone else is willing to purchase it for.