Correct, the mounts are interal to the speaker. The previous owned likely cut the speaker out and use the original bucket for the new speaker. Very very common.
You can get spacers for the front from Best Buy that will probably work. The ones from Crutchfield require you to mount your speaker to the adapter ring, then mount the ring to the door. Same as from BB, but the BB ones are easier.
If you have a crew cab, I don't think there is a spacer on the market that first properly (I didn't find one for mine). I custom cut a ring. The rear requires like 3/4" I believe to allow the speaker not to interfere with the glass.
The Polk speakers I installed came with an adapter ring, but is not typical of all speakers.