Ayako’s skin is sensitive to chemicals.
Very unnatural translation. The author from Tatoeba is not a native speaker (and I frequently found his other sentences also unnatural).
- The particle for the subject in this context cannot be が but は. 文子が～皮膚を持っています suggests that such a sensitive skin is a special gift, and 文子 is (the only) owner of the gift. The connotation doesn’t match the English sentence. Particle が functions as a selector for other competitors/candidates – が is for a vertical/hierarchical comparison. On the other hand, 文子は～皮膚を持っています means “other people probably don’t have it, but…” 文子 has a sensitive skin. Particle は is used for a horizontal comparison. There is no better/worse nuance. 文子 is just different from others.
- 「皮膚」を持っています sounds unnatural. We usually say 敏感肌（びんかんはだ）, 乾燥肌（かんそうはだ = dry skin）, 透き通った肌（すきとおったはだ = clear skin）etc. These phrases are widely found in cosmetics advertisements and in medical articles.
Also, TTS and the audio don’t match the English spelling. 文子 (a female name) can be pronounced as Ayako or Fumiko – both are correct and natural. But Ayako is more likely spelled as 綾子.
My alternative translation is:
Non-Japanese native speakers may find 肌の持ち主 not straightforward, but that’s the way we say in Japanese.