The only dual-type with normal in Gen I was Flying (until Fairy was introduced in Gen VI) which makes sense since they're based on regular, non-elemental birds.
Gen II introduced Girafarig, a Normal/Psychic type; this Pokemon had two distinct heads, only one of which had Psychic power.
Gen IV introduced Bibarel, a Normal/Water type which evolves from a Normal type; since the evolution retains a majority of the traits found in Bidoof, it retains its Normal typing.
Gen V gave us the Deerling/Sawsbuck line with the Normal/Grass typing. Like Girafarig, the Grass elements of Sawsbuck's design appear on it's antlers but not necessarily on the rest of the body. Also, Meloetta can be either Normal/Fighting or Normal/Psychic- in this case, the Normal typing represents it's musical theme, which may also explain why the Jigglypuff line retains it's Normal type.
Lastly, Gen VI gave us Diggersby, Pyroar, and Helioptile. Like Bibarel, Diggersby gains its secondary type through evolution. The Pyroar line have fiery manes but are otherwise normal in appearance, like Sawsbuck. The Heliolisk line is probably the biggest outlier here, but I think the idea is that it doesn't generate its own electricity (unlike most Electric-types) and therefore can't be considered a pure Electric type.
I'm sure I'm just stating the obvious here, but regardless I think its an interesting topic. In Bewear's case I think the Fighting type is secondary because it's a product of Bewear not knowing it's own strength, whereas Passimians typing is based on it's organization and training regime.
