In a multiwire branch circuit, can the two hots be connected to the same phase?
Yes, this does sound like a problem. Assuming your panel is like most, the breakers are on the same phase and you have two problems:
No Common Shutoff
MWBC is required to have common maintenance shutoff. That means if you turn off one breaker to work on the circuit, the other is turned off too. That is important for safety. This can be done with a handle-tie or by using a double-breaker. It is quite possible that a violation of this rule is what caused the problem in the first place! Without a handle-tie, and without realizing that the breaker is part of an MWBC, someone might move a breaker in order to accommodate other things, such as double-breakers for 240V circuits (dryer, oven, etc.). Once they do that, 50% chance that the 2 MWBC breakers will be on the same phase instead opposite phases.
When two breakers are side-by-side, a handle tie can be added. When the breakers are not side-by-side (with the special exception of the outer breakers of some quadplex breakers), you have to move at least one of the breakers to solve the problem.
Overloaded Neutral
Putting the two breakers on different phases (legs, poles, terminology varies and I'll leave it to the pros to argue that) means that the neutral will carry the difference between the two hots.
Putting the breakers on the same phase means the neutral will carry the sum of the two hots.
If you have a 15A circuit with 12A in use on each 1/2 of the MWBC, with different phases neutral carries nothing at all. With the same phase, neutral carries 24A, which is way too much for a 14 AWG wire. Neutral does not (normally) have a circuit breaker, so there is nothing to keep the neutral wire from overheating and starting a fire.