I'm pretty sure city states just aren't able to take other cities and, imo, they're specifically designed to only ever be one-city empires or they'd stop being "City States".
I've played about 150 hours now and have, several times, seen City States in a position to take an AI's city. However, they never have done so so that also points to it just not being possible.
If you levy their military then you gain full control of their existing units and they become your colour for a limited number of turns. I don't know exactly how many turns because I've only done it once and they reverted back to the city state before they made it to the other end of the continent to help me finish off the darn Chinese.