As with all definitions, there is no "proof" that the adopted definition is the right one but only a feeling that it better corresponds to our intuition.

In the case at hand, taking $R/IJ$ as structure sheaf would make union schemes nonreduced for no good reason. For example take $R=k[x,y,z], I=(y,z), J=(x,z)$. Geometrically you are describing the union $U$ of the $x$-axis and the $y$-axis in affine three space $\mathbb A^3_k$ .
It should have a reduced structure, correctly described by $I\cap J$, whereas $I.J=(xy,zx, zy, z^2)$ would make the function $z$ nilpotent but not zero on $U$, which feels wrong since $U$ should be a closed subscheme of the plane $z=0$.

A more brutal objection to the idea of defining the union of two subschemes by the product of their ideals is that a subscheme $U$ would practically never be equal to its union with itself, since in general$ I\neq I^2$ : we would have (almost always)
$$U\neq U \cup U$$

That looks bad!