This record appears to be an image copy of a page in a Manhattan borough death register. Unfortunately the person who shared it gave us no information about where he found this image. It may be a copy taken from the FHL film 447567 referenced in New York, New York City Municipal Deaths, 1795-1949, but it might not be; it could be from the Bodies in Transit films (see link below).
I would not automatically assume that any entries you found in the indexes at FamilySearch were indexed by people looking at this same image. Over time, registers got worn out and were copied, and you can't tell if there are multiple copies of a register or not unless you do some research. For Massachusetts, for example, an image might come from a copy of a town burial register (held by the town) or a state copy (produced to send off to the state). You'd have to look at the state statutes for New York to see if state law mandated that NYC make copies for the state, or whether it is likely that the only copy is the one in the NYC Muni Archives.
Let's assume for the moment that this is an image copy from the register held at the Archives and it is from FamilySearch's filming (which is likely). The entry you linked to at FamilySearch only gives us the FHL Film number -- we don't have any other clues in FamilySearch's Document Information such as the entry number that we could cross-check with the information found on the image itself. (Contrast the records from Queens in What New York City vital records are covered by this index on FamilySearch? where you can find more information once you open up the Document Information page.)
Searching for film number 447567 in the FamilySearch Catalog, the Film Notes for Manhattan death registers, 1795-1865 tell us that what is on the film and how it can be accessed:
Your extracted data at FamilySearch came from one of the volumes listed, is searchable (as we already knew) and can be accessed at a Family History Center. Clicking on the camera, I see that the film can only be accessed at the FHC and not at an affiliate library (such as my local genealogical society's library).
However, further down on the catalog page, it says "Another filming, 19 microfilm reels." Some of those films haven't been digitized yet (they only show a microfilm reel), but the volumes which overlap your film have been -- they have the camera with key icon, but no looking-glass, so they are browse-only. If you were looking at the image on film 447567 at your local FHC and the image was too hard to read, you could look at the other films to see if that filming was more legible.
We can see from the FHL's catalog that death certificates were not used in Manhattan until 1866. (See the collection: Manhattan death certificates, 1866-1919 ; index, 1868-1890 : Works Projects Administration, project no. 265-1-97-28; Work project no. 1 which is just a bit too late to be useful for your question.) But I would still look at the FamilySearch Catalog and at reference works to see what other records that could contain death information might be available. There is a span of time in New York City in the marriage records where records were produced in both formats, so 'always look' is my motto.
Also, you'll want to check whether the cemetery listed in this extract, Cypress Hills, is in Manhattan or in one of the other boroughs. Depending on the time period, and depending on state law, if the body was taken across jurisdictional lines, there might be a burial transfer permit (although few survive) and a second recording of the death in the jurisdiction where the burial took place. If Cypress Hills is not in Manhattan, check Manhattan bodies in transit, 1859-1894 (those films are also FHC-only) and the borough in which it is located.
You'll want to correlate this information with other records, too, so using reference guides and other finding aids like the United States Record Finder can help you find other records which might mention the date and place of death. Do a place search for New York City in the FamilySearch catalog, and check all jurisdictions -- Manhattan borough, New York City, and the Federal level -- to make sure you haven't missed anything of interest.
I encourage you to look further and to access a copy of the image yourself. When we look at image copies in isolation, we miss clues that might be on other pages of the register book. You can't properly evaluate the information in the shared document because you don't know its provenance. See Citing Data from Ancestry? for more on the perils of working from index data alone.
Record Finders and Inventories: