Just adding on to the solution guide by "footpickle" but not in a comment so it's easier to see.
In order to unlock this achievement you do need to use the Nano cores on the same brand of gear (ie. MG Gorgon). Doing this on a first run is a better option but I didn't like one gear set personally so I mixed up my gear and ended up not getting the achievement.
I'm now on my second playthrough and recollecting all the nano cores. I will update if you can leave some of the gear at level 5 and get the rest to level 10 just to get the achievement.
**Update**
I just re-collected all of the nano cores in the game and ended up at 18 for my second playthrough. I now know it does not count if you leave some of the gear at level 5 and others at level 10. What you can do if you don't like having a full set is upgrade all the gear of one brand, wait for the achievement to pop before closing out of the gear upgrade screen, and then quit the game. This worked for me so it should work for you.
I also just found out that you can get extra nano cores from defeating the big nanite enemies that appear. I'm pretty sure you have to hit them enough for them to spawn the mini-me enemy and then kill that one before you kill the main one. This only worked for me in NG+.
Hope this helps. Let me know if you have any questions.