Senator Gillibrand has also made a point of rallying against Republicans and courting progressive voters: she just received the endorsement of Indivisible, the liberal grass-roots group, which also endorsed Mr. Cuomo’s primary opponent, Cynthia Nixon, the actress and education activist. Ms. Gillibrand also ran on the Independence line in 2010 and 2012.

Mr. Schumer, the Senate minority leader and frequent foil of the Trump administration, has received between 100,000 and 220,000 votes on the Independence Party line in each of the last four election cycles, dating back to 1998, though the votes have never swayed the outcome of the actual elections. Ms. Gillibrand’s campaign declined to comment on their association with the Independence Party, as did Mr. Schumer’s office.

The Cuomo campaign and the Independence Party note dozens of other Democrats in other races have been endorsed by the Independence Party, including Anthony Brindisi, who is taking on Representative Claudia Tenney, in the 22nd Congressional District, one of Mr. Cuomo’s electoral targets.

The strange arrangement is permitted by the state’s “fusion-voting” laws, which allow candidates to collect votes on multiple party lines, even if the candidate is not a member of that party. This so-called cross-endorsing also serves a purpose for small parties, which must gain 50,000 votes in elections for governor to keep their ballot lines for another four years. And acceptance of the Independence Party line may simply be a matter of political strategy, of course: candidates never want to let their opponents scoop up votes on a line that they might occupy.

“The Independence Party is proud of the candidates we endorse,” said Mr. MacKay, the party chairman, noting they “select not from party labels but from the quality of the candidate.”

In a statement, Mr. MacKay also noted that “many of these officeholders have run with our endorsement for over a decade or more,” adding that most of those endorsed by his party win their elections.