I really have no idea if these are original or what. Evidently they have been reissued but the tube and packaging are very different. ... it would be cool if they were the real deal. ...
Yes, those are real-deal U.S. Tung Sol 5881's. They have all the hallmarks of the original tube. Nice!
... Now, if they were, would they be recommended over the 6L6s?
The original "5881"
is a 6L6; more precisely, a 6L6WGB (6L6, "W" - military designation, "G" - glass envelope, "B" - 2nd Revision after metal 6L6 and glass 6L6GA).
The 5881 has some ratings increases, and is designed to be mechanically rugged in a high-vibration environment (like in your bomb-sight on a B-52). It is rated for higher plate dissipation (23w vs. 19w for the 6L6/6L6GA), but that won't matter in the projector amplifier. So both tube types (as well as the metal 6L6) will work. Additionally, so will probably any other 6L6 variant, assuming they will physically fit in the space afforded.
By the way, I have found a schematic for it. It's a Victor or RCA Victor model 60B. ...
Can you attach the schematic here?
The speaker looks like it might be a field-coil type with the output transformer mounted on the speaker basket. If so, the speaker is an integral part of the amplifier's power supply, so the two should be used together. The schematic might help clarify that (or highlight that there's just some kind of line-matching transformer on the speaker).
NOTE: Eleventeen posted first, but we agree on all essential points. Separately, if you need the old-school metal 6L6, 6L6G or 6L6GA, I may have a line on a source for those for cheap prices. I doubt they can guarantee matching, though that may not matter in this case (I doubt you need screaming clean power from a projector amp).
EDIT: Photo of a Russian 5881 attached. Aside from the screened labeling on the tube base (which could be faked), note the lack of the 3 horizontal mica supports holding the top & bottom mica spacers whithin the tube. Also, while the plate is the same shape (which is unique to a 5881), note the Russian tube has notches at the top & bottom of the plate material. Your tubes (the real Tung Sol 5881) have no such notches.