So a doorbell transformer steps 120VAC down to a lower AC voltage. That’s what you’re measuring (surprised you can’t tell if it’s AC. Your multimeter should have an AC/DC setting).

The circuit puts the transformer, the button, and the bell (classically an electronagnet that accelerates the hammer) in series. When you close the switch (push the button) the hammer strikes the bell.

An electromagnet is just a coil of wire, and it is possible to pass a small amount of current through it without activating the hammer. If you’ve ever seen a doorbell with a light up button, the light bulb is placed in parallel with the button. That way it’s always on drawing a small amount of current. It also explains why the light switches off when you press the button. You’re shorting across the bulb, so it’s voltage is zero.

If you want to use this power for an electronics project, you’ll need to find a way to draw just a small amount of power from the transformer. It has to be very small or you’ll activate the hammer.

Drop a full bridge rectifier in parallel with the button, then a bunch of caps to give you a steady-ish voltage and then some kind of voltage regulator to make it useable.

Just remember that you can’t draw too much power, and you’ll lose that power whenever the button is pressed.

So when I installed my amcrest doorbell it came with a small component that jumps across the doorbell chime leads so that the push button and camera unit get continuous power. I didn't open it up but I assume it's just a small resistor. Here's the link to the manual. It's got some info on wiring too that might help. https://drive.google.com/open?id=1JmduLOXZetxixoFjSTGhf5IbDlLUHB7M

Smart switches/cameras tap into the power directly and actuate the chime circuit via a separate control (or use batteries). The ring pro does this and there’s a little power box that gets wired in at the chime unit for that purpose.

I’m not sure how it actually does the controls, however. A short term capacitor to run the door end temporarily and cutting the circuit to release a relay, maybe?

The easy way: take your existing wiring. Put an 18VAC to 5VDC power supply in parallel with the pushbutton. Then, the output of that power supply goes to the 5V USB input of your camera setup.

The downside is that it will reboot the camera every time someone rings the doorbell, because you are shorting across the camera's power supply.

You can put a resistor in series with the button to fix this. You will need to find a resistance that's low enough to still cause the chime to ring, but high enough not to disrupt the camera's power supply. Maybe start around 20 ohms. If you can't find a working resistor value, you can change the transformer to a 24V or 36V transformer, but make sure to keep that resistor high enough not to burn out the chime, and make sure your 5VDC power supply can handle the increase in input voltage.