What is the No Sim card icon for? On Android 7, when you pull down the top bar drop-down window (Quick Settings) a grid of 3x3 icons appears with tiles like Wifi, screen rotation on/off, Aeroplane mode etc.

On my new phone with a new SIM there is an icon called "No SIM card". What is this for?

I have a dual SIM phone with only one SIM card inserted, is it there to allow me to swap them? Is there anyway to get rid of it when I only have 1 card installed?
Here is a screenshot:
Since you have pointed out that you have dual SIM phone with only one SIM card inserted, so it is obvious that given icon is a Quick Setting tile for another SIM slot, which is empty in your case, hence No SIM card.
That tile works same as another tile shown there for your inserted SIM. It takes you to SIM cards menu in Settings
If you will insert another SIM card then it will be enabled.
You can remove that icon by enabling System UI Tuner and then hide that tile by drag and drop Quick Settings tiles.