Do planets and stars form the same way?

Answer 1

Actually, to quite an extent, they do. Whenever enough material in interstellar space accretes together until it has an appreciable mass, and thus attracts even more material to itself, you have the beginning of either a planet or a star. A planet if it is largely solids, or an amount of gas insufficient to make it a star, or a star if it is mostly hydrogen, and has enough of it to cause nuclear fusion at the center.

Answer 2

No, planets and stars form differently. Stars form from the gravitational collapse of huge clouds of gas and dust, achieving nuclear fusion in their cores. Planets are formed in the disks of material surrounding young stars, where small particles collide and clump together to form planetesimals that later accumulate to form planets.