I had read that paupers who died in the city received a "Christian burial", but what if the poverty-stricken person was Jewish or some other religion? Were they still entitled to a burial? Were efforts made to respect their religion?
Great question, in these instances burial would usually be done by a burial or benevolent society based on the person's religion. This was particularly true for Jews, who by the 1880s were becoming an increasingly large portion of both New York City broadly and New York City's poor. I cannot speak for other religions but in Jewish communities such work was generally done by organizations called "Chevra Kadisha" or benevolent societies which oversee Jewish burial for all Jews in a community, including the poor. These societies care for the body, cleanse the body, say certain prayers, wrap the body in a shroud and bury it. Though these were initially (and still are) primarily connected to synagogues or other Jewish communal bodies, in the late 19th century, particularly in New York and other major cities with high levels of Jewish immigration such as Chicago, these chevra kadisha began to be connected to mutual aid organizations called "landsmanschafn". These were groups organized around people who had emigrated from the same town or city in Europe, and created communal and at times political organizations around the shared experience of immigration. Many landsmanschafn operated burial societies, relief funds for families of the dead, burial funds, and even cemeteries. For example much of my family is buried in the Warsaw Cemetery in Chicago, which was operated by a landsmanschaf of Jewish immigrants from the Warsaw region. However, these organizations primarily focused on burial of the poor, but these organizations couldn not necessarily cope with the burial of the unidentified or those in need of a completely free burial
These burial societies were considered religiously highly important in late 19th century Jewish communities, particularly because Jewish burials have to be performed within a day or so of death, placing an additional level of pressure on Jewish burial societies that Christian ones did not have. Burying the dead is also considered particularly holy in Judaism, in death, all people are considered equal, a pauper's body is the same as a rich man's. With a significant number of immigrants, many living in crowded conditions with poor access to medical care and sometimes poor access to food, by the 1880s it became increasingly clear that burial of the impoverished dead needed to be somewhat centralized at least within the Lower East Side in Manhattan.
By the 1880s the main organization caring for the impoverished and indigent Jewish dead in NYC was the Hebrew Free Burial Association which is still in operation today. The organization relied on community donations to act as a chevra kadisha to care for and bury any impoverished, indigent, or unclaimed Jewish dead. Like in most religious traditions, a proper burial is considered a holy act in Judaism, so with the increasing number of impoverished immigrants coming the NYC in the 1880s in the wake of pogroms in the Russian Empire there was pressure both in immigrant communities and in wealthier Jewish communities to make sure the dead were cared for in a Jewish way. There was also a certain level of concern that if Jewish societies couldn't identify indigent dead, they would be buried as Christians, since that was the assumed religion in most circumstances in places like Potter's fields. The HFBA began in the Lower East Side, where most Jewish immigrants settled in the late 19th century and became well established by the end of the 1880s. At first they mainly buried children due to the high child death rate in New York's poorest areas, mainly due to infectious diseases. By the end of the 19th century the HFBA had buried over 15,000 people.
The organization became even more well known in 1911 when they were responsible for the burial of 22 of the victims of the Triangle Shirtwaist Factory fire, where most of the victims were young, Jewish or Italian immigrant women. The HFBA is still operational today and mainly serves the Russian Jewish immigrant community, and elderly Holocaust survivors, some of whom have limited familial resources.
"Dust to Dust: A History of Jewish Death and Burial in New York" by Allan Amanik
https://www.nytimes.com/2009/04/01/nyregion/01about.html?_r=1&ref=nyregion
https://www.hebrewfreeburial.org/who-we-are/history/