According to the NET translators notes:
This phrase may refer to an individual or to a church (or the church
at large). Some have suggested that the addressee is a Christian lady
named “Electa,” but the same word in v. 13 is clearly an adjective,
not a proper name. Others see the letter addressed to a Christian lady
named “Kyria” (first proposed by Athanasius) or to an unnamed
Christian lady. The internal evidence of 2 John clearly supports a
collective reference, however. In v. 6 the addressee is mentioned
using second person plural, and this is repeated in vv. 8, 10, and 12.
Only in v. 13 does the singular reappear. The uses in vv. 1 and 13 are
most likely collective. Some have seen a reference to the church at
large, but v. 13, referring to “the children of your elect sister” is
hard to understand if the universal church is in view. Thus the most
probable explanation is that the “elect lady” is a particular local
church at some distance from where the author is located.
A bible.org commentary further points out it is unlikely that the letter was written to an anonymous female recipient because St. John's third letter is made out to a specific recipient whose name is given (Gaius), so this would certainly not be a pattern for St. John.
Due to the fact that "elect/chosen lady" (ἐκλεκτῇ κυρίᾳ) is used with plural verbs in the majority of appearances (vv. 6, 8, 10, and 12) and the instances where it is not could still be understood as having a plural referent without a contradiction in meaning (vv. 1 and 13), it seems most likely that the "elect lady" is a collective reference, most likely to an unspecified church/group of believers.