Answer
The original statement is correct only under certain circumstances, e.g., if the 2 objects are made of the same material and have the same mass. However, if one of the objects is large or has a large specific heat capacity, it might only decrease in temperature by 1 degree while the other object increase in temperature by 10 degrees.
Increases and decreases in temperature are not necessarily equal.
It is true that the increase in internal energy of one object equals the decrease in internal energy of the other.