The one major difference with India is that the partition of Ireland was not meant to be permanent. During the Treaty negotiations in 1921 it was agreed that at a future date a Boundary Commission would be established to formalise the boarder between the Irish Free State and Northern Ireland (the Republic of Ireland didn't come into existence until April 1949). The Irish delegation was lead to believe that the Free State would gain a considerable amount of land, possibly two of the six counties. This would have made Northern Ireland economically untenable and would have resulted in unification.

However, when the Boundary Commission met in 1924 it recommended minor transfers to land from the Free State to Northern Ireland. Eventually there was an agreement that the boarder would remain as it was (and still is to this day) in return for the Free State making decreased contributes to the Imperial debt.

During this period the Protestant population on the island overall dropped. Wealthy southern Unionists, many of whom were attacked and had their houses burned, left the country, mainly for England and not the new northern state. There was more economic opportunities in Britain and they were as suspicious of a state controlled by Ulster Unionists as they were by a state controlled by Sinn Féin or its successors. Many well to do Protestants, especially in the legal trade, were essential for the continued functioning of the new Irish Free State and continued on with little disruption to their lives. Some Protestants fought in the War of Independence and the Civil War for an independent Irish State and saw it as being as much their country as the Catholics.

The Free State would become heavily Catholic with suspicion and discrimination against Protestants but there was no active campaign against them. Some historians, such as Peter Hart (Canadian historian of Irish history who died in a car accident a few years back, not the British military historian), have claimed that there was a campaign of ethnic cleansing carried out against the Protestant community during the War of Independence but this doesn't really stand up to scrutiny. Many loyal southern Unionists were executed as spies, especially in the Cork region, after giving information to the British. Southern Unionists who had served in the British army or who were known to be in contact with British officials were targets for suspicion and attack, not because of their religion but because of their loyalty. Religion often acted as a proxy for loyalty however and many Protestants with no connection to the British were held under similar suspicion.

On the other side Northern Ireland was gerrymandered to have an inbuilt Protestant majority and Stormont would later be described by Unionists as "A Protestant parliament for a Protestant people." Again, Protestant is being used here as a proxy for political affiliation. There was considerable violence in the region from 1920-22 and Catholics were viewed as a threat to the new state. Electoral districts were redrawn to give Unionists control over Nationalist majority areas and state forces took part in pogroms against the Nationalist community. Many did migrate across the boarder to the south but many others felt abandoned by the Irish Free State and either tried to get on the best they could in a country that could offer them more than the economically poor Free State could or saw their role as defending their people.

Neither state called for the removal of "the other side" or for "their own people" to migrate to their state. Northern Ireland didn't call for southern Protestants to migrate there and made no official attempts to eject the Catholics as they were still needed to keep the economy going. They were however held under heavy suspicion and repression was used to keep them in line. In 1937 Éamon de Valera changed the constitution of the Free State, removing all reference to the British monarchy and declaring that Ireland was a 32 county "pending the re-integration of the national territory". Calling for Catholics to move south would have damaged Ireland's claim to the region.

During the late 1960's however Nationalists in the Northern Ireland began to call for equal treatment. As mentioned, districts were gerrymandered and people were entitled to vote based on property ownership which overwhelmingly favored Unionists; only ratepayers and their spouses could vote and company owners got an extra vote. Nationalists began to hold civil rights marches calling for "One Man, One Vote". In response Unionist paramilitaries were formed to break up these marches with the assistance of the Royal Ulster Constabulary, the state police force. Nationalist housing estates were burned to the ground and the Irish Defense Forces began constructing refugee camps near the border in the Republic, one of which housed 6,000 people at one time. In September 1969 the Irish Government drew up plans to invade Northern Ireland, known as Exercise Armageddon. One theory is that they were hoping to force UN involvement in doing so. British forces were deployed to Northern Ireland to defend the Nationalist people and restore order; famously in 1971 they opened fire on a civil rights march and killed 14 Nationalists. This all created a massive humanitarian crisis which lead to mass migration to the Republic but still nothing on the scale of India/Pakistan.

Finally, in 1984 Margret Thatcher considered redrawing the Northern Ireland boarder, ceding Nationalist areas to the Republic and transferring populations between the ares, but this came to nothing.

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TLDR: "Catholic" and "Protestant" are often used as clumsy proxies for political affiliation in Irish history. While both the Southern Irish States and Northern Ireland would be dominated by a specific religious creed, individuals of the minority did not automatically view the other state as their home. Southern Protestants suffered discrimination in some state jobs and through a Catholic influenced legal system but were overall able to continue their lives as they had before partition. Some wealthy Southern Unionists who had suffered attacks (because of their political affiliation, not religion) left for Britain or other parts of the Empire as they had little in common with the Ulster Unionists who now ruled Northern Ireland. Many Protestants had played a part in the revolutionary and pre-revolutionary period in the South and would go on to have important roles post independence, such as Douglas Hyde who would serve as the first President of Ireland from 1938 to 1945.

Northern Catholics would fare much worse under Unionist rule but also felt abandoned by the Southern regime. The statement "The Republic of Ireland was for Catholics and Northern Ireland for Protestants the same way that India was for Hindus and Pakistan for Muslims" is wrong in that Northern Ireland was created to have a Protestant majority, not a near complete Protestant only population, and religion had little or no consideration in the creation of the Irish Free State. No attempt was made to summon the people of the states religion to them or to eject the people of the opposite religion.