Is the functionality of the breaker affected by the position in which it is mounted?
Yes and no, mostly due to the technology of the circuit breaker employed. For hydraulic magnetic technology refer to the diagram below. The effect of gravity on the tripping mechanism is due, or not, to the gravity forces being influential on the mechanism. For example, aiding the tripping will be faster and opposing tripping will be slower.
Circuit breakers using alternative technologies, thermal-magnetic and electronic magnetic, are not likely to be affected in a similar fashion.
The technology employed is not usually indicated on the body of the circuit breaker so that at any time there is the consideration to mount the circuit breaker away from the industry norm of vertical the manufacturer should be contacted to confirm if the proposed mounting has any effect.