There is precedent for your DM's position: command
The DM's position may be informed by the command spell, which says this
Grovel. The target falls prone and then ends its turn
The ruling is fair enough when taken at face value, given the use of that term in the spell's text, but the kobold ability itself does not explicitly require that the kobold fall prone.
You can make the (IMO reasonable) argument to your DM that the command spell's use of 'grovel' is a discrete magical effect that requires a spell slot to activate, whereas your kobold's ability is not explicitly a spell, nor is it a magical effect, and thus should not be treated as one. (It does not require a spell slot).
What is a spell (chapter 10, PHB)
A spell is a discrete magical effect, a single shaping of the magical
energies that suffuse the multiverse into a specific, limited
expression. In casting a spell, a character carefully plucks at the
invisible strands of raw magic suffusing the world, pins them in place
in a particular pattern, sets them vibrating in a specific way, and
then releases them to unleash the desired effect—in most cases, all in
the span of seconds.
Recommended ruling: kobold is not prone (unless you choose to drop prone)
While I can see a good case for either position, since prone is an explicit condition in Appendix A I find myself taking the "not necessarily prone" view on this. The DM may feel that because advantage is such a significant benefit to your allies that it takes something extraordinary (falling prone to beg, grovel, etc) to get that strong of a benefit in order fit their sense of verisimilitude.
Work with your DM to arrive at a common understanding.