I've seen it done on old stuff like 80s era alarm clocks, radios etc.
Probably two things:
1. You can avoid this in the first place by e.g. buying a right-angle display. (But if you can't find one, already available, you'll have a hard time doing this for less than 100,000/yr quantity...)
2. You can put more smarts on that little board, and wire it with a simple connector (like 3 wires for SPI, plus supply and grounds), then secure it anywhere in the box.
3. Or if things are still best with rigid mounting, use right angle header strips to mate them. (Can even use female socket strip to make it a connector rather than a soldered connection, if at some expense.)
As for assembly, I would think it would be okay, but:
a. You should probably have some sort of bracket to fix the angle during assembly. It might be a jig, used only during soldering (and thus cost some NRE and tooling charge), or a bracket that's screwed or clipped in (you can find such items standard e.g. from Keyelco at the usual distributors), then left in, or removed after.
b. You may have trouble forming a good solder joint. Ideally, such things would be mass-produced (where, for an assembler, "mass" might be more than a hundred or so?), which means wave soldering. You need to think carefully about the design of that joint.
There's probably two ways to do it:
You could do castellated holes. That would leave a relatively large gap between board edge and mating face. The 'barrel' (inside cylinder) of the cut-through hole faces the inserted board, and therefore leaves a lot of gap to fill. That's bad for getting solder to wick up. (You may need several tries to dial in the right radius and centerline, and the board fab may have issues with copper tearing off the milled edge -- that's always a danger with castellated holes, and why they add cost in tooling and yield.)
You could do an unplated slot. Copper has to run right up to the edge, and that forms a butt-joint with the inserted board. The copper edges have to mate very close (which is why the copper has to be right up to the routed edge), otherwise when the solder wave flows over it, it'll just wick off the small fillet that you need.
I'm pretty sure a big globby joint, like pictured, can only be done with hand soldering (or any of the really esoteric methods like laser plus solder wire/paste..). Fine for small quantities, but a huge deal in large quantity.
As for strength, you definitely don't have strength if the solder fillet is tiny (which you can expect from wave). Even with brackets, vibration might be enough to fatigue it away rather quickly! Conversely, even with generous hand-soldered fillets, you have the problem of delamination, because there's no plating wrapping around either PCB, it's all along one edge or another. You get the strength of the laminate under the fillet, but that, too, will fatigue and fail (under harsher conditions).
Oh, and obviously, you can't solder the top side, not with automated wave soldering. Putting solder fillets up there (as well as the bottom) would be excellent for strength, but awful for (automated) production. On the other hand, if you can make a plated slot work out, you'll be even better, because you get copper backed by laminate, plus whatever solder wicks along the whole joint!
Tim