This is one of the permanent oddities of Irish surnames: Teague (pronounced 'teeg'), Tague ('tayg'), McTeague, and Montague (as well as a few other variations) were considered to be the same name up into the early 20th century. You see some versions of this in other names, but rarely to this degree, and it's because Montague is a name elsewhere.

Teague as a name is an Anglicisation of Tadgh, a given name (pronounced like 'tiger' without the r). Mac Tadgh is 'son of Tadgh'. Mac an Tadgh is, literally, 'son of the Tadgh', wherein 'the Tadgh', like 'The Murphy', 'the O'Rahilly', and so on, indicated the head of the family. Mac an Tadgh is easily reduced to M' an Tadgh, which is then Anglicised to "Mon Teague". But when you stick that together, you get the entirely non-Irish Montague (almost, anyway), and almost is definitely close enough when it comes to writing down and copying out names. Once people moved abroad, to Australia or the US or wherever, this overlap of names was lost.

As you dig further into genealogical records of various kinds, you'll see plenty of instances where a father and child have different versions of the same name - Teagues inheriting from Montagues, Montagues on the birth certificates of McTeagues, and so on.

As for registration and documentation, there are... difficulties with that. Contrary to some romantic claims, most Irish people did have some documentation - at least at the parish level for baptism, weddings, and the like - but a lot of those records were lost in the Custom House Fire in 1921. This was part of the Irish War of Independence, during which much of the administration of the island of Ireland by the Local Government Board for Ireland was being conducted from the old Customs House on the quays of the Liffey. It was attacked and set on fire, and many, many records were destroyed, some dating as far back as the 16th century. Historians and genealogists have been unhappy with this ever since.