browserAction.enable()

Enables the browser action for a tab. By default, browser actions are enabled for all tabs.

tabIdOptional

integer. The id of the tab for which you want to enable the browser action.

Disable the browser action when clicked, and re-enable it every time a new tab is opened:

browser.tabs.onCreated.addListener(() => {
  browser.browserAction.enable();
});

browser.browserAction.onClicked.addListener(() => {
  browser.browserAction.disable();
});