Simplified version of @m-sarabi code.
func allows to return result, so no messaging is needed. Requires scripting, activeTab (+ contextMenus for menu) permissions

// background.js
  chrome.scripting.executeScript({
    target: { tabId: tab?.id || 0 },
    func: () => document?.getSelection?.()?.toString()
  }).then(result => {
    doWork(result[0].result)
  })