Is it possible for two different compounds to be made from the exact same two elements?

The answer is: Yes, it is possible.

For example, water and hydrogen peroxide.

Structural formula of water is H2O, while hydrogen peroxide has contains one more oxygen-H2O2.

Hydrogen peroxide is the simplest peroxide with the function of oxidizer, bleaching agent and antiseptic. It is more viscous and unstable than water.