We are using an SDE Geodatabase (10.3) with versioned data, however, when I look at the Versions in Tree View, it only shows dbo.DEFAULT (Grayed Out) and doesn't display any versions. Is this because all users are signed into the database under the same user (database owner) rather than having individual log ins?
Versions have nothing to do with logins.
By default, when a new geodatabase is created, there is only one version DEFAULT. Later you can create other versions under DEFAULT. The tree view will show only DEFAULT, if there was no other version created under DEFAULT. So please confirm that.
To add onto what Asrujit has posted above- there is an underlying difference between registering data as versioned, and the creation and use of child / edit or transactional versions in the geodatabase. The former prepares data for versioned editing through the creation of the associated database objects- primarily the delta tables (tables tracking adds / edits and deletes) for that (for example feature class) as well as entries in the required system tables. A quick tour of registering and unregistering data as versioned—ArcGIS Help | ArcGIS Desktop
The latter refers to the process of creating versions to perform edits in off of (or below / children of) the Default version which will always exist and is owned by the gdb administrative user. The link that was shared above is helpful for more information regarding this.
Put another way- you can have versioned data with only the Default version, and you can have transactional or edit versions created in a database with no versioned data... but if you wish to edit versioned data in a version other than default- you will need to do both (register data as versioned, and create and maintain edit versions other than default). Hope this is helpful!