Not every trinket "fits" in every location, for lack of a better term.
For the wall-mounted trinkets, there are two types of spots best described by using the layout familiar to people who use printers: "Portrait" spots (taller than they are wide) and "Landscape" spots (wider than they are tall).
"Portrait" spots on your boats fit both sizes of trinkets and you should have the entire set of ones you've purchased or earned available to choose from to place in those spots.
"Landscape" spots typically do not have a lot of vertical room on the wall (for example, the spot over the shelf opposite of the armory and ammo box on a sloop). You can only fit the "Landscape" sized wall trinkets in spots like that and the taller trinkets are not available to choose from for those spots.
Floor- or shelf-mounted trinkets are roughly the same. Some are larger, like the potted plants or the 'Forgotten Bosun' (my personal favorite) and can only be placed on a trinket spot on the floor of a room, while the smaller ones can be placed on shelves AND floors.
If a trinket is too large to fit in the spot you're trying to place it in, it won't appear in the 'trinket box' as an option for you to select even if you have purchased or earned it.