Thank you for the additional context! If you have a mac.com address, you’ve been an Apple user for some time. Me too!
Unfortunately, Gmail chokes on mac.com addresses. Fortunately, it’s easy to work around. Please try using your user name @icloud.com instead of @mac.com with the application specific password provided by Apple.
Some background, mac.com originated when Apple branded its cloud services as .Mac. Then Apple rebranded .Mac as MobileMe and we gained me.com addresses. Finally, Apple settled on iCloud and iCloud.com email addresses. Also, fortunately Apple had the foresight to map both mac.com and me.com email addresses to icloud.com email addresses. In other words, if your email address were firstname.lastname@example.org, then email@example.com goes to the same mailbox. You don’t need to tell others to stop sending email to you at mac.com, Apple takes care of and redirects everything sent to mac.com to iCloud.com in the background.