Hi,
Bear with me.
Once upon a time...
Apple created a messaging app called iChat. It worked with the AIM service only and could only Text Chat (so could the AIM set up at the time).
It swiftly added Video and Audio chatting at version 2 a short time later.
We are talking back at OS X 10.2 and earlier.
Apple have an agreement with AIM that Apple IDs (@mac.com names at the time) were to be recognised as valid AIM Screen Names. This continues to This day and then can be @mac.com, @me.com if linked to iCloud and @icloud.com as well
At iChat 3 in OS X 10.4 the Jabber account option was added.
This allowed you to use GoogleTalk as it was then to connect to Jabber Buddies as well.
Time passes...
At iChat 6 in Lion (OS X 10.7) Yahoo is added but is is text only and does not do Video and Audio Only chats.
OS X 10.8 sees Messages (at version 7) which adds the iMessages account to the fold.
As it pertains to your situation
This means that your Apple ID can be used to set up the iMessage account (normally the first step in setting up the App)
As this could also be your Google ID you can have a situation where your Google ID is used as the iMessages ID and the Google option ID.
In addition Google deemed that SSL (the method Messages uses) was not secure enough to login to their Servers (Strangely it dos not effect the Mail App).
To get around this there is as setting on your Google Account settings that "Allows Less Secure Apps" which means SSL Logins are allowed.
I have this set myself but I don't use Google's 2 Step verification.
Older Posts seemed to say you could not have 2 Step Verification AND Less Secure apps set at the same time. This may have changed.
So in Messages > Preferences > Accounts you initially see the iMessages option and the Bonjour option which is not active.
You cannot delete these two. You can only remove the Apple ID or untick the "Enable this Account" box to disable them.
After that you can use the + icon under the list to add other Accounts.
Yahoo in one option.
Google is another (it does not follow the Jabber Server naming format everyone else uses)
The "Other" option allows you to add other Jabber accounts or AIM (which is the same as AOL)
And there is of course AOL listed in the first drop down.
Other shows you this:-
With these options
AS you can see, behind the slide out I have lost of Accounts.
iMessages is at the top with mixture of AIM with the speech bubble icon, the AOL one that lists AIM as the type in grey text underneath that and Google and Yahoo.
Bonjour tend to be moved to the bottom.
If in System Preferences > Internet Accounts you have a Google ID it will tend to activate it for Messages if you upgrade to El Capitan.
If this is the iMessages ID as well that will also use SSL to login to Apple but it will use the Password that the ID uses at Apple.
To Start again remove (Sign Out) of the iMessages Account if it lists anything.
Highlight the Google account if you have partly set one up and it appears in the List then use the minus icon and delete it.
As you can see in the Pic above the Google Account is Highlighted and even though this in in Internet Account it has the buttons under the list. They are the same in Messages.
The Accounts pane in Messages should then show just the iMessages Account waiting for an Apple ID and the Bonjour account.
From your Post I am not entirely sure if you were trying to set up iMessages with a Google/Apple ID or actually setting up an Google Account in Messages.
9:31 pm Friday; November 13, 2015
iMac 2.5Ghz i5 2011 (Mavericks 10.9)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad
