CIV = Consular Immigrant Visa. While I cannot show an example to prove it, I believe this box serves the same purpose as the "Visa Number" column added to the passenger lists after the Immigration Act of 1924. The number entered is the number shown on the visa document itself, issued by the Department of State on Foreign Service (FS) Form 256 or 257 (one for quota immigrants, one for non-quota immigrants).
As usual, the number is no help in the research. This Foreign Service document number is not the USCIS Visa File number. The card I see is from 1943, so that visa originally went into a Visa File, though it may have later moved to an A- or C-file. If you have a 1945 manifest, the associated visa would have originally been filed in an A-file (because the LPR admission was after 4/1/1944).
Hope that helps,