Do I need USA-color-coded wire on fixtures with flexible cable?
Short answer: No
The National Electrical Code contains no particular color code for phase conductors. Brown is an acceptable color for the ungrounded conductor.
Green with a yellow stripe is used in a lot of equipment imported from Europe and is also acceptable to the NEC

250.119 Identification of Equipment Grounding Conductors. Unless required elsewhere in this Code, equipment grounding conductors shall be permitted to be bare, covered, or insulated. Individually covered or insulated equipment grounding conductors shall have a continuous outer finish that is either green or green with one or more yellow stripes except as permitted in this section. Conductors with insulation or individual covering that is green, green with one or more yellow stripes, or otherwise identified as permitted by this section shall not be used for ungrounded or grounded circuit conductors.

The blue neutral is also not a problem.

200.4 (C) Flexible Cords. An insulated conductor that is intended for use as a grounded conductor, where contained within a flexible cord, shall be identified by a white or gray outer finish or by methods permitted by 400.22.
400.22 Grounded-Conductor Identification. One conductor of flexible cords that is intended to be used as a grounded circuit conductor shall have a continuous marker that readily distinguishes it from the other conductor or conductors. The identification shall consist of one of the methods indicated in 400.22(A) through (F).
(C) Colored Insulation. A white or gray insulation on one conductor and insulation of a readily distinguishable color or colors on the other conductor or conductors for cords having no braids on the individual conductors. For jacketed cords furnished with appliances, one conductor having its insulation colored light blue, with the other conductors having their insulation of a readily distinguishable color other than white or gray.
The insulation shall be permitted to be covered with an outer finish to provide the desired color.

So, you could argue the fixtures are not appliances and therefore the neutral cannot be light blue but the last line says you can re-identify the light blue to white or gray.
So, it appears you can use the cords either way.
Happy day!