You are correct.
You have two choices, neither pretty.
Rewire the circuit
If you are in a rental unit, this option is not available to you without landlord consent; and then; a licensed electrician must do it.
Since the socket is NEMA 6-15, it is fairly likely the wire and cable behind it are 14 AWG. Open it up and have a look. If you're very lucky and it's actually 12 AWG wire, you can skip to the last step. Otherwise:
You will need to remove that wire/cable from there to the breaker panel and replace it with #12 wire or cable. This won't be so bad if it's in conduit, but if it's cable-in-the-walls, it may be prohibitive.
Once there's #12 wire in the walls, you change the receptacle to NEMA 6-20 and the breaker to 2-pole 20A.
Have them send you a different unit
NEMA 6-15 sockets are quite common for air conditioners. The power company should have gotten more clear on your requirements. I am surprised they made this mistake.
Contact them, say they blew it, and you need one with a 6-15 plug and socket. They should oblige.
Change the plug...?
Noting that the cable is inline on both ends, you could change both the plug and the socket to a NEMA 6-15 type. You can't leave the socket with a 6-20 because this would allow an appliance which requires a 20A circuit on a 15A circuit. Even if it doesn't trip the breaker, it'll run the circuit beyond the design intent. It breaks the whole system of coordination between Code requirements and UL listings that keeps appliances safe.