My cousin told me that New Zealand issued him a visa. Is it possible to obtain a visa from there if you don't have a passport?
Even with the NZ eVisa system, the answer to the OP's question is "no."
Applying for an eVisa requires that the applicant possess a valid passport. Like other online visa systems, the visa application is submitted online and requires the applicant to upload a photo of the applicant's passport information and signature pages. If granted, the visa is sent to the applicant by email, and usually must be printed and hand-carried at entry to the visa-granting country.
More information on NZ entry is available on this travel.stackexchange.com page.