As Adeese states, purchasing content on PlayStation 3 does not automatically unlock the content on PlayStation 4.
To elaborate on some initial confusion, you mention that the DLCs were available. While you have not stated as such, I assume the same is the case with the actual Destiny game.
Bungie ran a promotion where "Players who purchased Destiny before or on the 15th of January, 2015" were given a free upgrade to the next generation consoles format. This was to acommodate for the fact that players would not wish to purchase the game on both generations of console, and many had not yet made the upgrade.
This applied to Destiny, The Dark Below, and The House of Wolves. Having only played on next-gen, I can not confirm how this worked; however, when you log in to the PlayStation 4 copy, you are playing the same account as you were when you logged in to the PlayStation 3 copy. It is likely that this "upgrade to PlayStation 4 copy" has automatically applied to your account. This upgrade does not apply to The Taken King, and will not apply to any future content, unless directly advertised in future promotions.