Facebook Chat went live 2008-04-06. Google didn't view them as a threat in messaging but a threat in social networking; they thought they had messaging in the bag with their 2008-09-23 release of Android 1.0, and their 2008-11-11 update to Google Talk which brought voice and video calling, and their 2009-03-11 acquisition of GrandCentral, which was soon rebranded an invite-only Google Voice. But Facebook kept growing and growing and it had an integrated chat on a website where people went to spend their time, instead of Gmail, where they went to manage email.
To combat Facebook on social networking, Google launched Buzz with aggressive auto-opt-in on 2010-02-09. Buzz fizzled and attracted controversy for its aggressive piggybacking on Gmail, so Google tried again with Google Plus on 2011-06-28. That was a better effort, and it included the features "+Messenger", a text chat, and the video chat "+Hangouts". By this point Facebook had more than 700 million active users, and won messaging handily; its lead was cemented by the acquisition of WhatsApp on 2014-02-19, as Google continued to flail about.
In a post of mine last year [2] (which includes an older revision of the timeline linked in [1]), I speculate that it was Facebook Chat that killed the mid-2000s chat networks of old like AIM, Yahoo Messenger, and WLM, rather than Google Talk or any particular missteps of those incumbent chat networks. For example, I was surprised to learn that AIM was present in the iOS App Store at launch -- of course, there were no push notifications at the time -- not until 2009-06.