First: You cannot put a voltage higher than 3.3V into an analog in, so there is that. (I e, no hooking batteries straight to the analog in pin!)
Second: The VREF uses specific hardware to make sure the voltage is always at the fixed value, no matter what the input voltage is. It's like a voltage regulator. Trying to make assumptions about other voltages (such as 3.3V) based on the reference is not going to work well.
Third: The 3.3V voltage on the Teensy is also generated by a regulator, with low drop-out. As long as the input is about 3.6V or higher, it will generate 3.3V. Thus, you won't be able to tell much about the main battery in (USB or VBat) by looking at the 3.3V rail. Again, it's a stabilized regulator, so it's not expected to fluctuate much at all.
If you want to measure the supply voltage, the most straight-forward way is to create a resistive divider, and buffer it with a 100 nF capacitor. You can make the resistive divider quite high impedance to avoid draining the source, as long as you don't read it too often. The parallel capacitor will make sure the ADC sees a representative voltage.
This example uses a 604K upper resistor and 100K lower resistor, for a drain at 12V in of 17 microamps. The time constant for 604k*0.1uF is about 60 milliseconds, so if I read this no more often than every 500 milliseconds (half a second) it should be quite stable. If you're OK reading even more slowly, you can probably go to megaohms of resistance, although you start becoming susceptible to pick-up EMI noise at some point.
(FWIW, the circuit below generates 3.3V in to the analog input when the battery voltage is 3.3V*(704/100) or about 23.2V, so it's safe up to about 5S LiPo input. If you want to push it all the way to "absolute max" (which I think is 3.6V in the data sheet) you'll see 25.3 Volts so 6S probably won't kill it. If your actual battery is lower voltage, you can increase the lower resistor to gain resolution and lower leakage.)
Finally, if you want to measure the voltage, I suggest calling analogReadRes(12) to get a better resolution measurement.