What would FAA controllers be referring to when mentioning "code 0027?"
In the video the controller states that the target was a "code" 0027, which would indicate a transponder code.  Using ATC radar (in this case ARTCC) the controller can see what a secondary radar return (transponder) is squawking.  In this case apparently it's "0027" and displaying an altitude of 37,000 ft.

According to FAA Order 7110.66E entitled "National Beacon Code Allocation Plan (NBCAP)" code 0027 is assigned to ZAU (Chicago ARTCC) and is listed as IS-3 (code category) IS= Internal Departures - Special Use.

Here is the link 7110.66e

I have no clue as to why that transponder code is attached to that target in Oakland Center's airspace.