Seems like the guys on Wikipedia haven't come to a conclusion on this topic either: see Geodatabase on en.wp.
Definitely not. A spatial database is an RDBMS with spatial types, indexes and functions, see: PostGIS, Oracle Spatial, SQL Server Spatial, etc. A geodatabase is a particular database schema designed by ESRI for managing what they call "layers" as well as their bolted on handling of "versioning", "topology" and other ESRI application specific storage requirements. A geodatabase schema can appear in a spatial database, but also can occur in non-spatial databases like Access and even in custom file formats like FGDB.
I have to chime in and agree with @PaulRamsey on this. Geodatabase terms has become synonymous with ESRI technology. And sure while you might be technically correct that a spatial database is also a geodatabase I think it adds confusion to the issue.
What is the benefit, other than trying to prove a point, to calling them both geodatabases. And if it is to prove a point I think you only create confusion and make slightly less technical people confused.
Personally, for focussed Q&A, I would prefer to see questions about geodatabases tagged with the type of Esri geodatabase that is being used (file-geodatabase, arcsde, personal-geodatabase) but there seems to be a case for using esri-geodatabase occasionally.
To see what "geodatabase" synonyms are currently in place I am including a screenshot from one of the moderator tools. Switching geodatabase to point at esri-geodatabase instead of at spatial-database would be very easy to do.
My vote would be yes. I use them as synonyms and if I'm talking about a personal or file or sde geodatabase, I'll qualify it by saying an ESRI geodatabase.
I think most people who work in all-ESRI environments omit the "ESRI". Over time people assume that geodatabase is an ESRI format and use it as such.
technical note: Users with more than 2500 reputation and a total answer score of 10 or more on the tag, can suggest and vote on tag synonyms. Suggestions will be automatically approved when they reach a score of 4, and automatically deleted when they reach a score of -2. (from here)
Since at present there are only 3 users above the voting threshold and 1 just under, and very few tags with more than 10 uses by any single person, the mechanism can't work. So I suggest for the moment just propose the synonyms in this thread and the moderators can act on them when general agreement is voiced.