The trouble with these strings is that the LEDs are all in parallel. Sometimes they have a current limiting resistor(s), sometimes not.
I've seen one where the batteries fed the string via a single resistor. Others (particularly the solar versions with one battery) have a little PCB that steps the voltage up a little and also provides the current limiting.
In short, you can run these from a higher voltage, but the parallel wiring means the series resistor value needs to be chosen very carefully. Also, if the PSU voltage is too high, you waste a heck of a lot of power in the series resistor.
That said, the candy canes on my mesh display element employ parallel LEDs and I run them from a 5V controller with suitable series resistors.