Rooting already unlock phone without losing data, possible? As far as I have understood it's the unlocking part that requires wiping, and the rooting part doesn't. So, if I am already unlocked, can I root without losing my data? I've got a (red) Nexus 5 16GB Unlocked running Android 5.1.
Yes, you can root as many times you want without losing data if your device is unlocked. I've done that on my OnePlus One. 

For further clarification, it's the default mechanism in the device set by the OEM, that whenever the bootloader's default (locked) state is tampered it would issue a data wipe (a Factory Reset), most probably to protect the user's data from a possible compromise. 

Rooting on the other hand is moving a binary (su) under system (usually at /system/xbin and placing symlinks at locations like /system/bin) which with proper files permissions can be executed by an ordinary user  to gain supreme rights on the system.

It is possible that the rooting method you attempt in your device isn't the standard (via Recovery) one but needs a one-click tool, and some users have reported data loss with the latter, so you may take a backup to be on the safe-side. 

Further reading:


Bootloader States
Why does unlocking the bootloader wipe phone?
What does to a "to root a phone" mean?
How does rooting work?