I'm trying to install a brand new SHIMANO Deore FC-M590-L Hollowtech II crankset. I measured my bb shell to 68 mm and my old bb was also a 68 mm. The new bottom bracket is designed for 68-73 shells so I have to use spacers. As far as I can tell from instructions I have to use the three 2.5 mm spacers given in the package. Two on right and one on left side.

Problem is that left crank arm seemingly won't go all the way in. I'd say there's 1-2 mm of splines on left arm that is unattached to splines of the spindle. There is no room to push arms further in on either side. I must have hit this thing with a rubber maul like 50 times. I even broke the tip of the safety plate. Still won't go all the way in. Consequently when I try to finger-tighten the preload as much as possible there is still a couple mm of play on left arm. There is no play on right arm. What could be causing this? Should I remove the single spacer on left side?

Yes, remove the L sided spacer! Or, depending on chain line you could remove one spacer on the R.

What do you mean there is still play in the left arm?
You tighten the preload adjuster finger tight. Then torque the pinch bolts. After that you can move the non-drive side arm?
If the non-drive side arm isn't all the way one then the safety thing won't drop into the hole.
Shimano cranks are usually pretty fool proof, installed many and never had a problem.

Also, clean out the receptacle on the cranks. You'd be surprised the 'casting crap' that sometimes resides in them...

What the what? Stop working on bikes and take up golf (actually no, working on clubs is actually tricky)
Seriously though, there is only 2 things that can be going on here. Either the left crank arm is being prevented from fully inserting into the spindle, or your cups aren't spaced wide enough. Troubleshoot Danielson, troubleshoot.

There is play in left arm after preload, but before pinch bolts. I have not done the pinch bolts yet since there is still play. Safety plate is in the hole when 1-2mm of splines on left arm continue past end of splines on spindel. Is this normal and considered all the way on?

I have used all three spacers that came in the package. Shell is 68. Instructions clearly say I should use 3 spacers (2 right 1 left side). So I'm a bit nervous about removing one spacer but I'll try and see if there is still play. If anything the cups are spaced too wide. Otherwise left arm would be all the way on spindel? Or am I missing something? anyways I'll try removing the spacer and see.

Shell is 68mm. Crank is designed for 73mm shell. Spacers are 2.5mm. (73-68)/2.5=2. Use 2 spacers.

The 73mm setting uses one spacer. For 68mm you need 2+1=3 spacers.

Set the preload, tighten the pinch bolt and then check if anything is loose.

Left arms will have slop on the spindle splines until they are tightened, regardless of the preload. This is one reason that some will wear out the Bb prematurely, the preload was too tight being based on left arm feel prior to the pinch bolts being torqued down. Andy

I can second this. When I replaced my crankset with the same one you are, even with the proper preload I could make it wiggle on the left arm if I tried. As long as you are sure the spacing and such is correct, you should be able to set the preload, tighten the pinch bolts, and have no more wiggle.

Now I'm confused (Four!). You shouldn't be able to see the splines on the spindle since the arm should be against the bearing seal no matter how far they slide in. When installed, the arm doesn't bottom out against the spindle, it is somewhere along the travel with either arm pressed (ever so slightly) against the bearing seal. Wax on wax off.

Thank you all for the help. Problem solved. Solution was to test for sideplay AFTER tightening bolts. On youtube some videos say you should test for sideplay after preload and only move on to bolts if there is no play. This worked on my old FSA cranks but doesn't work on Shimano. I can also see now from videos that its normal to have 2-3mm of crank go past the end of right arm spindel. Cranks look good and run smooth. Lesson learned. For your information I can confirm that shimano cranks are strong enough to be hit with a rubber maul some 50 times no problem.