The way cylindrical button-top batteries such as AAA are made, is that they have a steel casing that goes the complete length of the battery and includes the negative terminal, this is all a single piece of steel. At the button top, there's an insulating washer or seal to the positive pole. Since for the lack of electrical connection, no current can flow through the battery, the battery probably does not significantly participate in the capacitive sensing, it's just the casing. By touching any part of the side casing, you are capacitively well-coupled to the negative terminal.
A capacitor is constructed from two conductive surfaces separated by a dielectric, an insulator. No direct current can flow, but high-frequency signal can pass through a capacitor, as the charge on one plate influences the other.
The capacitive sensor is a matrix of vertical and horizontal antennas placed every few mm across the touchscreen between the display and the cover glass - you might be able to see them in just the right lighting. They emit a high frequency signal and measure the current consumed. When there is no object in proximity of the screen, no current can flow. When your finger touches the screen, a chain of capacitors to ground or the environment is formed: the glass or plastic surface of the screen forms a first insulation layer and thus the first capacitor, and your body forms a capacitor to ground or the environment. Let's say you touched a battery casing directly, then it's just the extension of your body. But if you're touching it through the insulating thin layer of plastic that is glued onto the casing, you're simply introducing another capacitor. It is perhaps a matter of luck that the properties of the capacitor introduced still allow it to be effective at the sensing frequency, and even more lucky that it still works through paper - there is naturally also capacitance between the casing and the button top, but you may notice that the button top doesn't work nearly as well. This chain of capacitors to ground allows a small current to flow towards ground and then back as the polarity of the high-frequency signal flips over, and this current is measured by the sensor array.
The capacitive chain however has to go somewhere, to some fixed potential. You couldn't possibly hang a battery or capacitor against a touchscreen in a vacuum and expect it to work.