Irish Americans are extremely proud of their heritage because they encountered the same prejudice and abuse when they arrived in America but managed to overcome all obstacles in their path, ending up dominating cities and city governments, forcing their way into professions that did not welcome them (medicine and law), and creating their own education system to provide a pipeline to prosperity when alternatives remained bastions of anti Irish and anti Catholic prejudice. Catholic universities and professional schools remain very attractive third level iand fourth level institutions.
The RC parish system was the means by which communities were built, centered around the church and school. The churches were social centers, with musical evenings, talent shows, skits, am dram groups, men's and women's clubs, and charities providing food, clothing, job offers, and furniture for families down on their luck. They were little worlds unto themselves.. even today, if someone asks you in Chicsgo where you're from, the answer is the name of a Catholic parish, not a city neighbourhood.
Communities were built out in the newly settled lands of the Plains too - there are little towns along the east/ west railroads where you'll still find St Patrick's and St Joseph's parishes in the same town. These were the parishes founded by the Irish and the Italians, respectively. In cities, there are German, Polish, Bohemian,, Italian and Irish parishes, and intense rivalry between them. A mixed marriage was one involving individuals from different ethnic parishes... Suburban parishes tended to be more integrated as they weren't the first built, serving individual ethnic neighbourhoods.
Biden himself spent a lot of time with his grandparents when he was young, and would have heard a lot about his family's story of struggles both in Ireland and the US.
It's generally assumed that the US was always a place that welcomed the Irish with open arms, but it was a very hostile place initially. The Irish ( and other RC immigrants) stuck together because there was strength in numbers.