Grammaticality – ‘My Car Is at the Garage’ vs ‘My Car Is in the Garage’

As @snailboat pointed out, ears that listen is an example of synecdoche: a figure of speech in which a term for a part of something is used to refer to the whole of something, or vice versa. In this case, ear (a part of a person) is used instead of person; this usage is similar to saying the law instead of the police, or the church instead of the congregation.

Yes, you can say an ear that listens or a shoulder to cry on; nobody will take it literally, and think you want to talk to an ear, or cry on a shoulder, detached from the rest of a body.

As for being commonly used, the Corpus of Web-based Global English finds three sentences containing ear that listens, and three sentences containing ears that listen. This doesn't mean the phrase would not be understood.

We must be the extended hand that helps, the ear that listens and the compassionate heart that stands with them.

We need to approach our children not as character projects, but rather, we must see them with hearts of sympathy, with compassion and understanding, and with ears that listen.

It is an idiom. Another well-known combination of the same nature is "part and parcel". Both are used with singular verb forms.