windows.remove()

let removing = browser.windows.remove(
  windowId        // integer
)

function onRemoved() {
  console.log(`Removed window`);
}

function onError(error) {
  console.error(`Error:`, error);
}

browser.browserAction.onClicked.addListener((tab) => {
  let removing = browser.windows.remove(tab.windowId);
  removing.then(onRemoved, onError);
});

// in a script loaded by the page in the window
document.querySelector("#close").addEventListener(async ({ button }) => {
  try {
    if (button) return; // not a left click
    const windowId = (await browser.windows.getCurrent()).id;
    await browser.windows.remove(windowId);
    // this point will never be reached, since the window is gone
  } catch (error) {
    console.error("Closing failed:", error);
  }
});