Yes, the account may be flagged as no longer eligible for passwords. The error message from Google will tell you the reason.
Some thoughts on troubleshooting
can you login to the gmail account itself with the same username and password?
Try this: open a text document, type in the gmail username and password. Then open gmail in a private window in your browser. Copy and paste that info directly into gmail to log in. Did it work? Do the exact same thing with WP SMTP. In other words, copy/paste the information. Do not type it in and do not accept the defaults.
Sometimes a space can mess you up in the password field. Always count the number of * characters in the password field to make sure there’s no space.
Hope this helps.
Mike
I had the same problem. I was using this link to setup and noticed in the comments section:
HELPER GUY
Nov 29, 2015 at 9:26 pm
The above information is obsolete. As of 2015 you now have to use a SMTP plugin that is OAuth 2 compatible! I used Postman SMTP after trying for days to use WP Mail SMTP, WP SMTP, Easy SMTP, etc.
I have a paid Google Apps web domain and the option to “Allow Less Secure Apps” is no longer an option.
I installed Postman SMTP and am now able to send email.
Can confirm for any who stumble across here, @iamtheallspark is correct, and Postman SMTP works where everything else will no longer do the job.
https://wordpress.org/plugins/postman-smtp/