On Dashboard Tiles - is there a way for a user to have their own tiles that is only visible to them and not everyone in their role or is it only filtered by role?

If you are talking about Portlets that you see in the home page, they can be personalized for each person.

What I am referring to is the Dashboard Tiles module. I do not see an option where a user can select which tiles to show. All I see is the option to apply a role(s) to a tile or when adding the Custom Portlet to select Dashboard Tiles - but no option for the user to select which tiles to add.

@pen one the Tiles will show if they are permitted for that Role. You would need to give the a unique set of permissions such that only your Target user sees the Tile.