This looks like a potentiometer with a built in on-off switch on the back.
When you either push on the knob or turn it to the end of it's range does the resistance between A and B change from infinite to 0 ohms? If so, you have confirmed that it is a switch.
The resistance between 1 and 3 should be fixed and between 1 & 2 (or between 2 & 3) vary as you turn the knob.
These are (were) commonly used as combined volume controls and on-off switches on radios before the digital age.
In response to the comment.
Since you measured an exponential change in resistance you have what is known as a logarithmic taper potentiometer. The other type is a linear taper. Logarithmic taper is used for volume controls since human hearing does not respond linearly but rather on a decibel scale.
If you measured across 1 & 2 instead of 2 & 3 you would have perhaps found the switch as "on" when the potentiometer is "on" as you put it. Looked at another way, the resistance across 1 & 2 will increases as the resistance across 2 & 3 decreases. In fact the resistance between 1 & 3 should remain constant as you turn the knob. Also for any place you put the knob, if you add up the resistance between 1 & 2 with the resistance between 2 & 3 that total should equal the resistance between 1 & 3.