Add database to availibility group I have two replicas in availability group. Also I have additional instance instance3, with dummy databases that are not in availability group.
As time has passed, some new things started to be stored on that dummy database and now I have to add it to Availability group.  

I followed user guide as described here:
https://docs.microsoft.com/en-us/sql/database-engine/availability-groups/windows/availability-group-add-a-database
but when I get to specify new database, in wizard, that I want to add to availability group, I don't see that database in the list.  

Did I miss some step in the middle or I am doing it completely wrong?

Additional note is that there is a database with same name but from another instance in availability group. Will that maybe create problem? I guess there can't be two DB with same name and now is little late to change name of DB.
It sounds like you are trying to add a database to your AG from a node other than the primary (is instance3 even part of your cluster), you can't do that.  You would have to fail over to instance3 and then add it.  You could also take a backup of your database, restore it to the primary, then go through the steps to add it again (you'll have to do a full backup and log backup from the primary).

Your biggest problem is going to be the name.  You are correct, you cannot have two databases with the same name.  It will have to change or you'll have to move it to a different cluster.