Area of a circle with radius $r$: $A_c = \pi r^2$ has the units $[A_c]=\text{rad}\text{L}^2$. Area of a square with side $a$: $A_s = a^2$ has the units $[A_s]=\text{L}^2$. Since $[A_c] \neq [A_s]$, we shouldn't be able to add the two.
If we are able to add $\text{L}^2$ and $\text{rad}\text{L}^2$, why aren't we able to add radians and steradians together?
As pointed out in the comments, "$\text{rad}$ is a unit of angle for which a circle makes an arc length equal to the radius of the circle; steradian $\text{sr}$ is the unit of solid angle for which an area equal to the square of the radius of sphere is projected onto the surface of the sphere. If we don't assign any units to angle, why are we not able to add radians and steradians together? $\text{sr}$ is effectively $\text{rad}^2$, I think."
This is indeed a topic of ongoing discussion amongst researchers, for (a few) references:
I could go on, but I recommend searching Google Scholar for keywords such as "Is rad an SI unit?"
As another reference, NCERT refers to $\text{rad}$ and $\text{sr}$ as units in "Units and Dimensions" (Chapter 2, Class XI physics textbook), albeit dimensionless ones (which is also a topic of ongoing discussion.
