In ArcGIS ModelBuilder is it possible to use a %Name% field value from iterated features in a query and, if so, what syntax should I use?
Yes. You'd use the inline variable in the exact same way you're using it in your output featureclass.
Inside your Select tool, build up your expression something like:
'"the_field" = \'%Name%\''
However -- I question your overall workflow. You're iterating over FeatureClasses and getting the "name" of the featureclass. Your workflow as-is means the Name of the featureclass is going to be used to "select" a row (matches an attribute) in another featureclass. I suppose that could be valid, it just seems unlikely.