When we say "solve for the variable", we mean to solve the equation because the solution of the equation is finding the unknown variable.
So, the similarity between "solving for a "variable" to "solving an equation" is that both actions imply looking at the value of that variable.
Similarity: both actions imply looking at the value of that variable.
On the other hand, a slight difference would be in the case of having several variables in the equation. In this case, both actions could represent slight differences, because if we have three different variables and we say "solve for y", that means we must isolate that variable, and the result is just an expression. But, if we say "solve the equation"