The FSN, (Federal Stock Number system) was in use from 1955 to 1974.
It is always possible that your Data plate is not original. That's the reason there is no FSN. If the data plate became lost, or unreadable, then a new one is ordered, and the unit is suposed to fill it out as best as possible. Normally, in the old days, such information was kept in the Log Book. Something else that often came up lost or "improved".
"Improved". I wont say I ever had any such experiences, personally, but I may have heard a story, or two. There are unscrupulous people in the world. Sometimes they decide to take things, from one place, and deposit them in another. When such things happen, some people would remove identifying information, such as data plates. When such things happen, it becomes difficult to trace ownership, or properly identify the equipment. So when it is discovered that something has been misplaced, and recovered, a unit has to still account for items. Then, a new data plate is ordered, and what info that is on hand, is put on the data plate. If this happened in the Army, the AR, (Army Regulation) covering this situation says something to the effect, "If the gaining unit is unable to properly identify the FSN, or Serial number, one must be made up". A number would be made up, and stamped on the plate. A note is then placed in the logbook and everyone is once again happy.
Doesn't mean that's what happened to your forklift. But it could have. Also, perhaps the Navy bought some equipment to use, for a short term, and after the project was done, the equipment was reissued to some other unit. And, since control and accountability is a VERY high priority, so someone ordered a data plate, and filled in the info that they had, and then riveted it to the forklift. That way, when a commander walks around doing his 10% inventory every month, he had something to look at.
Another scenario, is that the Navy did not want to buy a lot of these forklifts. They also did not want to go to the expense of assigning a FSN, and most importantly, did not want to go to the expense of producing TM, (technical manuals) for a low density item. When all the services agree to procure the same equipment, these costs were split between the services. And the costs are not inconsequential.
Could you shoot a few more pictures, and give us a bit more detail? The military had about a kazillion different types of 4K forklifts. maybe if someone can see a bit more detail, they might be able to give you a better idea, what one it MIGHT be.