It certainly seems like the same Niels Peter Sorensen.
> "Carl who changed his name to Nielsen and we have no idea why."<
It could be that Carl, like Niels, had a father named Nielsen. If he was born illegitimately he could be given his mother's maiden surname. In America, instead of keeping his mother's maiden surname, he may have chosen to adopt the surname of his biological father. This is only a guess based on the back story for Niels Peter Sorensen.
a NIELS PETER SORENSEN
found in Hadsten, Sabro, Aarhus, Denmark
1880 Danish Census
Aarhus, Sabro, Hadsten, et huus, Hadsten by og sogn, , 9, FT-1880, C6867
living in the household of Kristen Henrik Jensen, skomager (shoemaker)...
Niels Peter Sørensen 17 Ugift (unmarried) skomagerlærling (shoemaker apprentice) born here in this parish <-------
an ANE SEVERINE SORENSEN
also found in Hadsten, Sabro, Aarhus, Denmark
1860 Danish Census
Aarhus, Sabro, Hadsten, Hadsteen By, En Gaard, 52, FT-1860, D6177
...working on the farm of Peder Nielsen Schjødt...
Ane Severine Sørensen 28 Ugift (unmarried) Tjenestefolk (hired worker) born in Haldum Sogn, Aarhuus Amt <------
Aarhus, Sabro, Sabro, Sabro By, et Huus, 49, FT-1880, D5016
Søren Sørensen 43 Gift Husfader Indsider born in Laasby Sogn, Aarhuus Amt
Ane Severine Sørensen 48 Gift hans Hustru (married, his wife) born in Haldum Sogn, Aarhuus Amt <--------
This Danish emigration record gives the last place of residence for Niels Peter Sorensen as
Hadsten sogn, Aarhus. This matches the 1880 census for Niels Peter and for Ane Severine Sorensen that I posted.
DANISH EMIGRATION RECORD
Navn: Sørensen, Niels Peter
Kontrakt nr.: 262500
Fødested: ? (birthplace not noted)
Sidste oph.sogn: Hadsten [last place of residence]
Sidste oph. amt: Århus
Sidste oph.sted: Hadsten, Aarhus
Bestemmelses land: USA
Bestemmelses by: Boston
Bestemmelses stat: Massachusetts
ELLIS ISLAND IMMIGRATION RECORD
Passenger Manifest, line 0046
Niels Peter Sorensen, age 30, bootmaker
Ethnicity: Denmark, Scandinavian
Last Place of Residence: Denmark
Date of Arrival: Aug 05, 1892
Port of Departure: Copenhagen
So the next step is to find the brother's birth record in Denmark to establish the connection.