- I'm coming to the conclusion I don't look my age, and I don't act my age. (source)
- In his cowboy hat and boots, he certainly looked the part. (source)
In both of these, look is a complex-intransitive1 verb, with my age and the part functioning as predicative complements (PCs), not as objects (Os).
The criterion we can apply to both is exchanging the noun phrases (NPs) with adjective phrases (AdjPs) (I'm coming to the conclusion I don't look goodAdjP; he certainly looked handsomeAdjP) or bare role NPs, as they are both admissible as PCs, but not as Os.
Another criterion, which isn't sufficient, but is necessary, is that a PC can't be the subject of the corresponding passive clause (UNGRAMMATICALmy age is looked by me; UNGRAMMATICALthe part was looked by him), which an O could be.
I'd regard both expressions as idiomatic in meaning, but I don't see why the above analysis wouldn't apply anyway.
1 a term used in The Cambridge Grammar of the English Language (2002) by Huddleston et al., whose analysis I've tried to follow in this answer