Can one Gmail account have a second email address?

Not a second @gmail.com one, no. If you want a properly separate gmail address you have to make a new account. But you get free aliases already. Add a plus and any word after your username (yourname+shopping, yourname+bills) and mail to it lands in the same inbox, so you can hand those out and filter on them. Dots are ignored too, so your.name and yourname are the same address. If you own another address somewhere else, Settings > Accounts and Import > Send mail as lets you send from it inside Gmail.