If atoms are made up of smaller particles, such as protons, neutrons, and electrons, then why are atoms considered the smallest particles of an element?

Because all protons are the same, all neutrons are the same, and all electrons are the same, and all atoms of all elements are made of them, so the smallest particles of every element are the same particles.  It's only when some number of them get together and make an atom that the properties of an element begin to show up.

Let me try it like this, and see if this does anything for you:

All buildings are made of brick, cement, wood, glass, pipes, wires, etc.
But all bricks are the same, all cement is the same, all glass is the same, etc.
Houses, stores, shacks, skyscrapers, and hunting lodges are all made of them.
You don't know what kind of building you have, until some brick, some cement,
some wood and some glass all get put together.

Similarly, you don't know what kind of atom ... of what element ...you have, until
some protons, some neutrons, and some electrons get put together