Then do I need a current-limiting resistor when connecting two 3V LEDs
in series to a 6V supply voltage?
It's not enough information.
Your '6V' nominal supply has a tolerance, and so does the '3V' forward
voltage on the LED. If the LED were a perfect diode, it'd double its forward
current every 15 mV or so, and 6.15V would, while being
nominally still '6V', force about a thousand times the intended current
through the LED.
Your LED choice, and battery choice, includes internal resistances
and temperature coefficients, and measured-value tolerances. LED
direct to battery CAN work, but you have to examine all the vertices
(i.e. all the extreme values of temperature, voltage, etc.) and not
just one point which is the 'nominal' set of values.