what would be the behavior of an object that was immersed in a fluid if the object and the fluid had the same density

The object will remain where it was placed on the fluid

Explanation:

If an object is immersed in a fluid, with both the object and the fluid having the same density, the force of buoyancy (the force acting upward) will be equal to the gravitational force (the weight pulling the object down) making the object remain in the position it was placed in the fluid. Meaning it will neither sink down or float through the surface.

For practical purpose, it should however be noted that the chances of this happening is very low.

NOTE: When an object is less dense than the liquid/fluid, it will float and when it is more dense than the liquid, it will sink