Gerald Bray, James, 1-2 Peter, 1-3 John, Jude, 231-232:
Clement of Alexandria (Adumbrations, i.e. fragments on Comments on the Second Epistle of John) believed the elect lady was an actual woman, named 'Elekta', and was identical with the 'she who is in Babylon' from 1 Peter.
Hilary of Arles (Introductory Commentary on John 2) said the elect lady was a personification of a church, as described above.
Andreas (Catena) decided the elect lady was a literal woman, some sort of leading figure in her church or household.
The Common Critical View
According to Bart Ehrman The New Testament: A Historical Introduction to the Early Christian Writings, 186, 'most scholars' view the elect lady as a personification, similar to Hilary of Arles above.
Birger Olsson, A Commentary on the Letters of John, 61:
The letter is addressed to "the elect lady and her children" (2 John 1), and it ends with a greeting from her "elect sister's children" (2 John 13). These formulas are unique, although they do have counterparts in the milieu. In antiquity, a group of people, such as a city, could be referred to as a "lady," kyria.
Olsson provides a handful of examples from the Hebrew bible, where the Israelites are called 'daughter of Zion' and 'virgin Israel' (Zephaniah 3, Jeremiah 34), and the Samaritans are 'Jerusalem's sister' (Ezekiel 16, 23). Olsson continues, 62:
The nearest linguistic parallel to this is found in 1 Pet 5:13, where "she who is in Babylon, chosen together with you"—that is, a congregation in Rome—greets "the chosen" who are named in the letter's introduction (1 Pet 1:1).
Bart Ehrman, The New Testament, continues:
In the course of his letter, however, the author stops speaking to this "lady" and begins to address a group of people ("you" plural, starting in v. 6). This shift has led most scholars to assume that the term "elect lady" refers to a Christian community, a group of people who are considered to be the chosen of God.
Beyond that, the letter does not provide enough information to determine which church this would be.
The author has often been identified as John the Elder, who was closely associated with churches in Asia, particularly Ephesus. The overlap between the Johannine letters and the letters of Ignatius and Polycarp in vocabulary, and theological and ecclesiastical concerns1, especially in such close temporal proximity,2 substantiates that association.
If we are correct to identify the author, 'the Elder', with this figure John the Elder, it is possible the churches personified as the 'elect lady' and her 'sister' were churches in the Asian region sometime at the turn of the century.
1 Ehrman, The New Testament, 190. The Johannine epistles are critical of 'antichrists', a group of (from the Elder's perspective) of pseudo-Christians who deny Jesus 'has come in the flesh'. This was evidently an emerging theology of docetism. The letter of Polycarp of Smyrna to the church in Philippi quotes 1 John 4.3, and further alludes to 4.7. The letters of Ignatius of Syrian Antioch repeatedly emphasize the 'flesh' of Jesus, and Ignatius' letter to the church in Smyrna specifically criticizes 'certain unbelievers' who claim Jesus 'only seemed to suffer, as they themselves only seem to be', and will be condemned to be 'divested of their bodies, and be mere evil spirits'.
2 The New Oxford Annotated Bible (ed. Michael Coogan). The Johannine epistles 'were written ca. 100 CE'. Ehrman, The New Testament, places Ignatius and Polycarp 'around 110 C.E.'