Does every psychiatric drug mimic a neurotransmitter?

Not exactly. Psychiatric drugs work in a number of ways. Most affect brain systems that increase or decrease a particular neuroregulator, e.g., blocking a receptor or slowing reuptake back into the releasing neuron. Others are mimics (agonists) that work by fooling the brain into thinking a system is more active than it is.
Psych MEds. Psych meds are designed to increase or decrease neurotransmitters depending on the condition, they may work or may not work its 50-50 usually.