Opps, I got carried away, and bought a bunch, thinking how great this would be for breadboarding. Which it is for a couple of LEDs. Which it is not if you wanted a byte, or more, of LEDs. I measured the current draw separately of 5 LEDs. The red ones draw about 20 mA at 5 volts and 10 mA at 3.3 volts. The other colors measured about 15 mA at 5 volts and 6 - 8 mA at 3.3 volts.
So, if you were using an Arduino nano for example, you could use about 5 LEDs total if all on at the same time, but not six. You could still use them by sticking in a current limiting resistor, but that’s the point, not needing that.
The resistor in the LEDs is designed to protect the LED, not what the LED is connected too.
A potential solution to your issue might be to write your code so that you cycle through multiple LEDs very quickly. That way only one would be on at a time but to your eye, multiple would appear to be lit.
Not sure what your limitation is. The nano has a 40mA limit per pin, either driving or sinking. You should be able to drive as many LEDs as you have IO pins if you are using a 5V nano (limited by your USB power supply). If you are using a 3.3V rail (not sure if a nano has this), you will be limited by the voltage regulator, which is probably be a couple hundred mA. And that thing would get toasty. But as TS-Chris said above, you need to have a current limiting resistor in series with your LED. ~300 ohm is fine. If you just go from IO to ground with your LED, you might not immediately kill your LED, but you are making both the LED and Arduino die a little on the inside because they are becoming the current limiter.