I'm only showing 1.5 V across the E and C when the Arduino writes LOW.
What is going on?
When the Arduino writes LOW the transistor is turned off, so it is not shorting between the Collector and Emitter.
However this is only true when the voltage between Collector and Emitter is positive. If it is negative by more than ~0.5 V the transistor will turn on in 'reversed Emitter follower' mode. If an AC voltage is applied then it will act as a half wave rectifier and the meter will show the average rectified voltage.
"But I am not applying an AC voltage!", you say. But you probably are. Your body is effectively a large conductive plate with a typical capacitance to ground of ~100 pF. Voltage will be induced onto it from the electric fields put out by mains wiring. Depending on the strength of this EMI, just holding onto the meter lead could induce several volts onto it.
Normally you don't notice this induced voltage because the coupling capacitance is very low, so at mains frequency its impedance is very high. Therefore - unless the circuit you are probing also has very high impedance - the coupling capacitance will drop almost all the voltage due to its higher impedance, and what's left will be below the threshold for rectifier action. Without rectification the multimeter will show close to 0 V because it is designed to reject AC on DC ranges.
The circuit below represents the situation.
simulate this circuit – Schematic created using CircuitLab
Here's the result of simulating the above circuit in LTspice. The average voltage of this waveform (which the meter would read on DC) is 1.5 V.