What's the difference between 眼熟 and 面熟? I've always assumed that you use 面熟 for people because they have a face (面). However, I recently heard a native speaker use 眼熟 about a person.

Are the two phrases fully interchangeable?

If not, what are the subtle differences (which I'm apparently missing)?
They are not fully interchangeable.

In 面熟/脸熟, 面/脸 are nouns for 'familiar what'. 面熟/脸熟 means 'familiar face'.

In 眼熟/耳熟, 眼/耳 are functioning as adverbs for 'how is it familiar'. 眼熟 means 'looks familiar', 耳熟 means 'sounds familiar'.

As a result, 面熟/脸熟 are mostly used for people, 眼熟/耳熟 can be used on anything that you can look at or listen to. For a familiar face, either one is fine.