I went to the lab where the pinch whacker and berry charm trinket are available to pick up, but I saw you cannot craft nor obtain multiple copies of these items. I know various trinkets that drop from candy shards and/or bugs can have multiple drop rates. I am curious to see if anyone knows if certain trinkets like the thor and berry charm trinket are available as multiple copies for each player in a multiplayer world? I cannot seem to find much information regarding the trinket system and the drop rates of specific ones, but I just want to see if anyone knows if the more rare ones are available for each player or not. Thank you!

So far someone manage to find 20 trinkets. 
I do believe that the trinket you mention that have drop rates are for the multiplayer experience. 
But I have found 2 Trinkets that look the same, they look like badges and my friend and I both understood that the shared world would have some limit to our gameplay.
There should be 5-6 Badges as trinket, these trinket are for sure only obtainable once. So it up to you and your friend to decide who get to use which one.