There is no real consensus on the role of a COO in startups. His or her responsibilities change from company to company.
I myself held this position for about 15 years in the two startups I co-founded. The main characteristic I give it is its catch-all scope.
A bit like the last rule of a firewall or a web server configuration that processes the remaining cases after all the preceding rules have been applied. The COO takes care of everything that is not taken care of by someone else. Then he or she delegates portions of his or her responsibilities as managers are appointed to handle them.
At least that's how I've experienced it, and the conversations I've had with other co-founders who have worked in this position seem to confirm this description.
I've selected a few interesting articles for you to get acquainted with, or learn more about, this role.