I am familiar with the OSM plugin, however I mainly use the "Download, Extract, and Symbolize OSM Data" tool.
When I use this, I enable "Extract OSM Tags into Standalone Attributes." The feature classes that are output have a large number of fields joined representing multiple attributes (most are Null), and I am able to label roads or other features using the "name(tag value)" field.
I suspect that your tag values (including name) will be in the SDE.small_osm_relation table and you would need to join these based on OBJECTID or OSMID. If this isn't the case, perhaps you could try "Download, Extract, and Symbolize OSM Data" tool in this plugin, as that has given me good results in the past (however it does limit the extent you can download).
In the attached image, you can see what the field I see looks like (I dragged it from far right)