Are five conductor wire runs (two neutrals) for two circuits still considered a "multi-wire circuit" for code purposes?
In this case, is the run still considered a multi-wire circuit for NEC purposes?


Short answer: No

Here is the relevant Code definition:


  Branch Circuit, Multiwire. A branch circuit that consists of two or more ungrounded conductors that have a voltage between them, and a grounded conductor that has equal voltage between it and each ungrounded conductor of the circuit and that is connected to the neutral or grounded conductor of the system.


Since you would have two hots and two neutrals they are now separate branch circuits. They do not have to be on a two pole breaker or have a handle tie since they do not share a neutral. They do not have to be on opposing hot legs, they can both be on the black (left) or red (right) feeder bus.

Good luck and stay safe!