Sorry, I chose a wrong value for y. My conjecture is that there is a single solution for all y between a and b, except when y == Sqrt[a b], which is the limit of the right-hand side as c goes to +Infinity and to -Infinity. But I doubt that there is a formula for the solution.

Check the syntax: Solve does not take Assumptions, you give them with the equation itself. What values of a,b,c did you use? With a = 2, b = 4, y = 2 I get no solution:

With[{a = 2, b = 4, y = 2}, 
 Solve[y == 2^c a (4 + 4 (a/b)^(3/c))^(-c/3), c, Reals]]
With[{a = 2, y = 2, b = 4},
 Plot[y - 2^c a (4 + 4 (a/b)^(3/c))^(-c/3), {c, -4, 4}]]

Reduce gives False:

Reduce[y == 2^c a (4 + 4 (a/b)^(3/c))^(-c/3) && a < y < b && a > 0 && 
  b > 0, c, Reals]

Anyway, with respect to c the equation is tascendental, very unlikely to have a symbolic solution with symbolic coefficients.

I have similar issue with other equation. I have no idea how to solve it. It works fine for predefined values of a,b,c, however I get empty list as output for exact solution.

Solve[y == 2^c a (4 + 4 (a/b)^(3/c))^(-c/3), c, Reals, 
 Assumptions -> a < y < b && a > 0 && b > 0]

Any idas how to deal with it? Thank you.