Personally, I think our name is Sarah is not unacceptable, though it sounds strange based on the infrequency of such an answer. The truth of the matter is that grammatically, they only have one name between them: It is a singular subject.
Consider:
- Our horse's name is Grimm.
- Our address is 615 Gramercy Park Drive.
- Our son is fifteen.
- Our cheesecake is the one covered in chocolate swirls.
In these examples, two people share the relationship with the subject, but the subject is singular.
I think if there were only 10 names to choose from, our name is would quickly become standard, as no one wants to say I'm named Sarah, and so is she several times a week for her entire life.
If the subject is plural, then it needs to follow its proper grammar rules:
- We are both named Sarah.
- Our horses' names are Grimm and Reaper. The Black one is Grimm, and the brown one is Reaper.
- Our sons are 15 and 18.
- Our addresses are different, as we have separated.
If there were only 20 names to choose from in the world, I'm sure the construction would be quite common and acceptable.
Our, us, we can always refer to (addressee is the person spoken to):
- the speaker and and the addressee
- the speaker and a third party
- the speaker, the addressee and a third party
Some languages have different words for an us that includes the addressee or not.
In the case of "our family", it is not strange that someone would refer to their family in that way. I could tell you, or a stranger, about "our family", meaning "my family", not yours:
In our family we always married inside the family: my father married my mother, my uncle married my aunt, and my grandfather married my grandmother.
My family would surely also be correct, but some speakers prefer to use our.
I have heard people refer to our house in the same way. It is as if the speaker is not only speaking for themself, but also for the rest of the people involved:
We, members of the Johnson family, think our family is a good family. Our family is great!
We, the owners or occupants of that house, like our house a lot. Our house is lovely!
Even when only one person of the group is present, they can speak from that perspective, meaning exactly the same:
I think our family is a good family. Our family is great!
We like our house. Our house is lovely!
The same thing happens when your friend tells you about his football team:
Our team is doing great! We won the last five matches!
That does not mean you are part of that team!
Best Answer
"Our bodies" refers to a collection of bodies owned by a group of people including the speaker. If a group of people share only a single body, then "our body" would be correct. When speaking of an actual physical body with an immune system, it's extremely rare for it to be occupied by more than one person.
"Each of [our|your] bodies" similarly refers to that same collection of bodies. As such it is still plural; even though it is encouraging us to consider every member individually in sequence, the end result is that we are considering the entire collection.
However, if you're referring to "the human body" as a generic construct as opposed to a group of specific people and their physical incarnations, that is an acceptable time to use the collective possessive form "our" and still use the singular "body":