A hard-to-categorize use of their. Clearly, their is co-referential with each other and shares its reciprocal meaning, so that A knows B’s phone number and vice versa.
On the face of it, this looks like a so called picture NP, where each other’s is a possible antecedent for a reflexive as in (i). Compare (ii), which shows that the antecedent can’t be the subject – I think this is most people’s judgement, but there is some variation.
(i) The children saw each other’s pictures of themselves.
(ii) *The children saw Bill’s pictures of themselves.
The problem in the cartoon is that each other should not be able to bind the possessive. So in picture NPs we have the following data:
(iii) *Bill saw Mary’s pictures of her. (with Mary and her co-referential)
(iv) *Mary saw Bill’s pictures of him. (with Bill and him co-referential)
Co-reference is possible with the subject, however:
(v) Bill saw Mary’s pictures of him. (with Bill and him co-referential)
(vi) Mary saw Bill’s pictures of her. (with Mary and her co-referential)
Back to the couple in the sofa. The co-reference isn’t actually between each other’s and their, then, but rather between we and their. So, their has a plural antecedent, but it’s first person, rather than third. I’m not sure everyone would approve of this particular usage, but for those who do what it appears to show is that they, them, their and themselves are used in quite a wide range of contexts and with quite a range of antecedents. Singular they is just one case. The case here is ‘imposter they‘ or, since the antecedent is first person (exclusive, i.e. the lady in the armchair is not included), ‘exclusive they‘.