I guarantee the voltage do add up. No matter how you wired the circuit it will not defy the laws of physics.
If you have another volt meter (or scope) measure the voltage between VD and GND while measuring the voltage across the LEDs. See theory below.
At 23 mA the voltage across each LED will be about 2.0V.
Source: LTL-307EE Datasheet
The voltage across the 10Ω resistor will be 0.23V, totaling 4.23V from V_SUPPLY to VD. This would leave 11.77V across the input and output (VD to GND) of the TLC5925.
When the TLC5925 is not enabled there is a leakage current of about 1 µA.
Source: TLC5925 Datasheet
The minimum Vf of each LED is about 1.5V.
The voltage across the 10Ω resistor will be about 0.00001 V.
This would leave 13V at VD.
If you plan to use more than 4 outputs of the TLC5925 you will need to increase the value of the resistor. With your current circuit you are dropping 0.27 watts on the one output.
At 23 mA a 400 Ω resistor will drop almost 10V. This will not have any effect on the current. The current will continue to be set by the TLC5925. This will lower the voltage across the TLC5925, and alleviate thermal stress on the TLC5925 when (if) more outputs are used.
And how does LEDs behave in this condition - when led driver is turned
The LEDs only need the minimum Vf to turn on. LEDs do not have a minimum current. Vf will follow the the datasheet's IV curve. Some LED datasheets (e.g. OSRAM) will show the Vf at very low (µA) currents.
The Rdynamic of the two LEDs is about 3 MΩ at 1 µA.
When you put the meter across the LEDs some current is going to flow through the meter. This will reduce the current flowing through the LEDs which will increase the LEDs Rdynamic. When the Rdynamic increases more of the current will flow through the meter. This repeats until all current is flowing through the meter and none through the LEDs. Similar to thermal runaway.
The voltage is not likely still 3.0V but the meter may not be reading the voltage due to the current flowing through the meter rather than the LEDs. The voltage may be anywhere between 16V and 0V. If you have another volt meter measure the voltage between VD and GND while measuring the voltage across the LEDs.
Instead of measuring directly across the LEDs try measuring between the LEDs to GND. The voltage between the LEDs to GND should read 14.5V.
This is only a theory. You would have to run some empirical experiments to verify.
If you do not have another meter, try adding a 10 MΩ resistor in series with the meter's lead. What happens to the voltages?