Totally depends on the razor - some razors have round pins, which likely slide less than slotted heads. On my Rex Ambassador with horizontal slots (definitely a premium razor), there is some perceptible slide, but less than the Feather (I'd guess less than a half-milimeter or less either way). On my Taiga (also premium - horizontal slots) no movement at all. On my Muramasa ("T"-shaped slots), almost imperceptible movement. On my ATT Windsor (with small pins), probably about the same as the Feather. So - it depends. And some movement is not a bad thing - likely built-in to ensure a fit for almost any blade, no matter how imprecise the blade cutout, because I'm guessing some blades are made with less precision than some razors. And again, all of my razors get a visual check when inserting the blade and before shaving, and I appreciate the very slight leeway to be able to align everything perfectly (also OCD ).Yes, this is exactly what I'm talking about, including your description of "...slides back and forth...".
It does bother me (probably my OCD) and I do exactly what you say -- which is manually make sure that they line up perfectly as I tighten it up.
I was wondering whether this is true of other premium stainless steel razors or if they have even tighter tolerances.
Shaving is an art - with some science thrown in - but very much an eye-hand coordination, learned physical activity requiring practice, skill and finesse. If there was a perfect razor, perfectly machined, with perfectly fitting blades, perfectly aligned, and our faces and beards were identical, we'd all use the same razor and shave the same way, and never need to check anything. What would be the fun in that?