Multimeter not showing same current as bench power supply
It's probably a bit of offset in the amplifier for the current measurement. You should believe the DVM before the power supply current or wattage display.
You can't expect metrology-lab measurements from the display on a $99 programmable power supply. The display has 0.1% F.S. error at that point. The power is no doubt simply calculated from the voltage and measured current. They may have biased it that way to prevent the more disturbing display of current when there is none.
I would expect it to be more accurate as you get into the hundreds of mA to A. You can verify by increasing the voltage slowly from zero, you will likely see the power supply current display read zero until the DVM is reading around 5mA, then it will go up 1mA for every 1mA.
Even inexpensive DVMs tend to have a very good auto-zero strategy and will always read close to 0 with 0 in. Inexpensive ones may have poor span accuracy so the reading will diverge from true the higher you go away from zero.