Rugby , hockey, and cricket have an all- Ireland team. Only the Commonwealth Games sees Northern Ireland compete on its own (and necessarily, since the Republic is not in the commonwealth), so why is football different?
In 1921, There weren’t that many clubs from what is now the Republic of Ireland (RoI) in the Irish League (this was affiliated to the Belfast-based Irish FA, which ran the game across the island), but those that were there had been growing increasingly resentful of the Irish Football Association’s perceived Belfast bias.
A huge problem for the now-RoI based clubs was that Gaelic Football was increasingly popular and at that time the GAA clubs did not allow their members to play “foreign” sports such as football (rounders, however, was ok) and the IFA was perceived to be doing little to help them. Struggling financially and for players, the two Dublin clubs, Bohemians and Shelbourne, struggled to compete.
In a troubled political climate in 1921, Glenavon from Lurgan in the North and Shelbourne of Dublin fought out a tense draw in the Irish Cup, forcing a replay. This replay was meant to be held in Dublin, but at that stage the Irish war of Independence was in full swing. The IFA moved the game to Belfast, ostensibly for security reasons, and Shelbourne were incensed. Along with Bohemians and other non-League teams, the clubs left the Irish League to form a new league, the Free State League, in June of that year. The Football Association of Ireland was formed in September of that year, when the Free State League merged with the Leinster Football Association, and claimed jurisdiction across the whole island.
In December of that year, the Anglo-Irish treaty came into effect and the Free State became a country in its own right (though with the King still as its head of state, a full Republic did not come into being until 1949), albeit with a somewhat painful birth.
The Belfast-based IFA continued to select southern-based players, while from 1936, the South started to pick northern players. In those days you were not tied down to one country so there are many players who played for both. Confusingly, both sides continued to call themselves “Ireland”, a situation rectified in 1954 when FIFA forced them to accept their “Northern” and “Republic” monikers, by which point both stopped crossing the border for players*, and this is the situation today.
The rugby union, on the other hand, had no such issues and decided not to split on independence, and nor did other sports, causing the situation we see today, where we have an Irish Football Association, a Football Association of Ireland, an Irish League and a League of Ireland. One advantage of the two teams though is that football fans don’t have to sit through the cheese fest that is “Ireland’s call”.
*Due to Northern Irish citizens’ ability to get an irish passport if they so desire, Northern Irish footballers may opt to play for the Republic, but not vice versa. This situation is somewhat political, but is actually the same as in other sports. An athlete may choose to compete in the Olympics for Ireland or Great Britain.