Let's simplify your question with correct terminology: you have a presynaptic neuron and a post-synaptic neuron. The presynaptic neuron can release multiple signals (neurotransmitters) into the synaptic space, or cleft. The postsynaptic neuron can be equipped with multiple sensors (receptors) that are specific to detect certain signals.
The signal and the sensor don't have to match, although this is costly if it's poorly tuned. Most neurons that are entirely unresponsive are pruned. Darwinian selection makes circuits much more efficient than what pure chance alone would produce, but remember that nature can be quite imperfect and stochastic.
However, you can have neurons that have a dopamine sensor (receptor) and provide an acetylcholine signal (neurotransmitter) and vice versa.
Acetylcholonergic and dopaminergic neurons can be connected. Here is an example.
Also, most people forget that neurons not only form chemical synapses but electrical synapses as well. That means the circuitry does not rely on ligands and receptors.