I noticed the same thing when changing f150 prop and got a little nervous. I did a Google search and found many guys saying it is normal so I stopped worrying. It is actually the prop shift that has in and out play. Doesn’t matter how tight the nut is.
I would be willing to bet the play is not in the prop. If you remove the prop. you will most likely find that the hub shaft is what has the bit of play in and out of the gear case.
As mentioned is the play the prop or the shaft. I questioned the shaft play on my Yamaha 150. It was within spec so I forgot about it. Like Eric mentioned, if you have the wrong thrust washer, that will cause you problems