You're right: 1.13 custom crafting doesn't provide NBT tags for more customization (unfortunately).

How to test for items, fun question! It's actually not that much different from 1.12, well, "sorta" because as you probably know the /testfor command got shishkababed :D

On the good side though is that we got something much cooler in return!

So to try and keep bias out of my comment (apologies for the sillyness: I'm a good mood right now ;)) I shall concentrate on a randomly chosen 1.13 item: the Turtle Shell Piece.

Obviously Mojang did something wrong there because this should have been the "Shell shell" :D

Now, the really cool part about 1.13 is that we can actually combine several command aspects into 1 command. So....

give @p[nbt={Inventory:{id:"minecraft:turtle_shell_piece",Count:1b}]}] minecraft:turtle_shell_piece{display:{Name:"{\"text\":\"Shell shell\"}"}}

Dunno about you but this excites me. Unfortunately this isn't good enough. See, the problem is that once you gave a renamed shell you cannot easily take back the vanilla one. I can specify what the id or Name should be, but cannot negate that.

SO... Let's apply a little more 1.13 awesomeness.

Regular command block (but you'll probably want to change this to a repeating one eventually):

tag @p[nbt={Inventory:{id:"minecraft:turtle_shell_piece",Count:1b}]}] add hasshell

So now we marked the player who has a shell in his shell, errr: inventory :)

Chain command block, always active but it doesn't matter too much if you make this conditional or not (I suggest conditional):

clear @p[tag=hasshell] minecraft:turtle_shell_piece

So now we took back the item, next stop: replacement.

Chain command block as before (I suggest conditional):

give @p[tag=hasshell] minecraft:turtle_shell_piece{display:{Name:"{\"text\":\"Shell shell\"}"}}

This will give the replacement.

Finally, another chain command block:

tag @p[tag=hasshell] remove hasshell

And that will clear things up.

So when they drop the item you want to run a setblock command. hmm..

I may get back to you on that one. You'll need an /execute command.

/execute as @e[type=item,nbt={display:{Name:"{\"text\":\"Shell shell\"}"}}] at @s run setblock ~ ~-1 ~ minecraft:diamond_block

Yah, that no workie. Nor this:

/execute as @e[name="{\"text\":\"Shell shell\"}"] run say m000

This does work but seems wrong:

/execute as @e[type=item,name="Turtle Shell Piece"] run say m000

(if you pick it up it'll have the renamed name).

I think I hit a bug. I'll think this through and get back to you :)