The 2-pole 20A breaker is correct for an MWBC.
The only difference between a 2-pole breaker and two 1-poles handle-tied is that the 2-pole breaker guarantees common trip, and the handle tie does not.
Yes, a 2-pole breaker will let you flow 18A on one leg and 19A on the other. The breaker has no connection to neutral and does not know or care what current is flowing on neutral.
MWBC does not require common trip, only common maintenance shutoff. Nonetheless, we recommend a 2-pole breaker, because they are available everywhere for 2x the cost of a single... rather than 2 breakers with a handle tie, because listed handle ties are often difficult to find and add unnecessary cost, and we don't want you using a nail or nothing at all indefinitely while "get a handle tie" sits on your D-list of priorities. 2-pole is also the only reliable advice when double-stuff breakers are involved.