I am doing a practice problem about buoyancy force. If there are two objects, (one made of wood and one made of metal), and we hold them underwater, will the buoyancy force of both objects stay the same? That's what I believe at least.
Yes the buoyant force is the same. The buoyant force is defined as the weight of the water displaced or $\rho V g$, where $\rho$ is the fluid density and $V$ is the volume displaced. Because the amount of water displaced is the same for each object (I assume you are stating that they have the same volume), the buoyant force is the same for both.
The buoyant force will be same if the fluid density is the same and the volume of liquid displaced by both objects is the same. The formula to calculate upward buoyant force is ρ × V × g , where ρ is the density of the liquid the object is immersed in, V is the volume of the displaced liquid and g is the gravitational acceleration.
I have created this Buoyancy Simulation which will help you understand the concept better.