The technical term for this is 'clitic doubling', i.e. the doubling of the article. The mechanics are not entirely understood (at least by my understanding), but it has to do with the information structure of the sentence. With an object initial sentence like this it can occur, but also with verb initial sentences like τα έχει τα κλειδιά ο πατέρας μου and subject initial like ο πατέρας μου τα έχει τα κλειδιά. These are grammatically correct, but in practice clitic doubling doesn't occur uniformly across all the different word orders. The important thing is that the doubled clitic goes in the position directly before the verb.
Clitic doubling is related to the notion of resumptive accusative, a type of prolepsis (anticipation). I'm not surprised it's in modern Gk (MG) because ancient Gk had prolepsis (Smyth p. 488). Mackridge writes about this in his Modern Gk Language, p. 223: "An important function of the clitic pronoun is its proleptic and resumptive uses. When used proleptically, the clitic pronoun anticipates the object pronoun....when used resumptively it recalls an object which has already been stated." I think we have resumptive clitic here. Mackridge then gives examples and states that "a clitic pronoun is always used when όλα ('all') is the direct object." He has quite a bit more to say about the topic. It's a little bit like anticipatory indirect object pronouns in Spanish, such as les mando dinero a nuestros hijos, "I am sending money to my kids," which causes difficulty for English speakers who find the indirect object pronoun unnecessary, redundant, and so we forget to put it in when speaking Spanish. We'll have a similar problem remembering to put in the second τα in this sentence. My question is (slightly different from the one below): Will this sentence work with the clitic inserted proleptically? ο πατέρας μου τα έχει τα κλειδιά.
To answer a question stated by some commenters, this phenomenon almost never occurs in the formal speech and written form of Greek. However, they are very common in the vernacular and they are used, as far as I can understand, for emphasis/focus. But the two τα must not be considered the same. There is not a full explanation of this, but I believe that the first τα is the article accompanying κλειδιά, while the second τα is the shortened version of the neuter plural form of the personal pronoun αυτός, αυτή, αυτό (pl. αυτά, i.e. they - neuter). Which means that literally this sentence would be: "My father has them (the) keys." Example: Έχεις τα κλειδιά; Ναι, τα έχω (τα κλειδιά)! (Do you have the keys? Yes, I have them (the keys)!)
I appreciate your comment here. According to Mackridge (Modern Gk Language, p. 223), "The neuter singular τό [sic] (sometimes the plural τά) is often used to refer to a whole clause." He gives an example of τά 'μαθες; Πέθανε ο Μπρέζνιεφ. Ναί, το ξέρω. "Have you heard (it)? Brezhnev's died." "Yes, I know (it)." In this sentence, he says, τα "perhaps refers to τα νέα, 'the news,' while το ...is optional." His comment seems to back up your point that a full explanation of this phenomenon is hard to come by. He doesn't say anything about τα as possibly being a shortened form of αυτά. Horrocks has a section on clitic pronouns and syntax on pp. 108 ff. in his Greek: A History of the Language and Speakers (2nd edition). Apparently, he also wrote a seminal article on this topic in 1990, "Clitics in Greek: A Diachronic Review" in Greek Outside Greece II (Athens) pp. 35-52. I haven't read that article, but in the book (p. 109) he writes that classical Gk 3rd person anaphoric pronouns, αὐτόν, 'him,' etc., function effectively as clitics...they are, of course, the source of the modern clitic pronouns τον etc., via the reduced forms ἀτόν etc. that are sometimes attested in low-level texts of the Hellenistic and Roman periods." He corroborates your hunch.
The second τα is redundant. Is it necessary? Yes, the phrase looks truncated without it for a native speaker. So, Object, τον/την/το etc, Verb, Subject. A reverse form Subject, Verb, Object, that is ο πατέρας έχει τα κλειδιά does not have τα. In this case the syntax Subject, Object, Verb is odd, and not used. Notice that this kind of Syntax is permitted in Greek, but not always, except for some poetic speech phrases, which have freedom in the Syntax.