The Collection system provides stat bonuses for characters who add specific items to it.
For items such as Profession materials, your character is only registered to have "collected" the item if they loot it themselves from a slain monster. Items received in trade from other players, transferred between characters via account storage, or purchased on the Marketplace are not counted towards your collection progress.
Some players have reported that they own certain mounts, but that the mount is not registered in their Collection. This is due to an error. Please send a support ticket and we will send you another copy of the mount to use, which should register it properly. We will have to confirm that your character already owns the appropriate mount before we can credit you with another mount in this manner.
Some players have reported that certain pets are not added to their Collection once used. There are multiple versions of some pet items, and certain pet items are of a version left over from before the Collection system was added. Before trading/purchasing a pet with the intent of adding it to your Collection, check the pet item's tooltip and make sure that the appropriate Collections are listed on it. If the Collection names are not present, the pet item will add the pet to your list of owned pets, but it will not count towards Collection progress.