The user manual of your multimeter tells us why:
In the current measuring mode (DC or AC): input impedance approximately 3 kΩ.
Which in all honesty is just a joke.
So with 5 V the most current you can get is 1.67 mA, but it doesn't even tell us the range of the input impedance, so the value you measured is "fine".
It says to be able to measure up to 4 mA (which is another joke), you need at least 12 V to get close to the measuring range by creating a "short" with your test leads.
Actually we can calculate the input impedance for your meter to be: 5.12 V / 1.367 mA - 15 Ω = 3730 Ω.
A good meter has something in the range of 10 Ω or less, depending on the selected range. The µCurrent, for example, has 0.02 Ω for mA measurement, 10 Ω for µA measurement and 10 kΩ for nA measurement.