Should the bottom of a shower enclosure be watertight?
Yes, a shower base should be sufficiently water tight to contain an inch or so of water above the drain.  I know that mine does.  Yours doesn't appear to have a lip to contain the water, so that door needs a gasket to both contain the water and prevent damage to the door mechanism.

It sure looks like your builder forgot a seal or chunk of rubber gasket.  If it's recent work, I'd definitely ask about it.

And sure, you want to have your drain running right, but all drains eventually clog.  A working drain doesn't mitigate an unsealed shower base.  You'll need it one day regardless.