Dishwasher-proof way to personalize plastic lunch containers
Both types of plastic, polyethylene and polypropylene are "slippery" and do not take glue or other adhesive attachments well. By extension, paint products are likely to slip off in some manner.

It would be more permanent to create a "damaging" effect to the plastic in an area that is not structurally critical, a section that doesn't provide for sealing or for strength in handling.

It appears that the corners of the lids may qualify in this regard. You could use a soldering iron or wood marking iron to engrave your mark(s) on the surface of the plastic. If you used sufficient caution and did not puncture the surface, you could mark almost anywhere on the container.

The flange around the rim of the bowl (redundant?) is another non-structural location. Even if you chose not to mark initials or names, a bump or divot or similar depression would provide identification unlikely to be duplicated by another owner.