It’s suspect to just say that this wave of immigration will turn out fine just like the past waves because there are so many differences between then and now.
Perhaps the biggest difference between then and now that just destroys the ‘we’ve been here before’ argument is that all of the past waves came to an end! Congress passed a series of restrictive laws in the 1920s that cut off mass immigration, and then followed over 40 years of low-moderate levels of immigration. It was during this time (complete with the unifying experience of WW2) that the Irish, Italians, etc were brought into the mainstream of American society.
There is nothing similar on the horizon. This current wave will not end on its own. It will take action from Congress, and there is little hope of that happening. And so long as annual levels of immigration remain high it’s a safe bet that immigrant communities will remain solidly Democratic.