This is not a weird case, but a Leontief production function which is not homogeneous of degree one, but homogeneous of degree $b$. You can see this if you use the connection between a C.E.S. production function and the Leontief one.
Consider
$$Q_b=[a K^{-\rho} +(1-a) L^{-\rho} ]^{-\frac{b}{\rho}},\;\; b>0$$
$$\Rightarrow Q_b = \frac 1{[a (1/K^{\rho}) +(1-a) (1/L^{\rho}) ]^{\frac{b}{\rho}}}$$
Take the limit when $\rho \rightarrow \infty$. Since we are interested in the limit when $\rho\rightarrow \infty$ we can ignore the interval for which $\rho \leq0$, and treat $\rho$ as strictly positive.
Without loss of generality, assume $K\geq L \Rightarrow (1/K^{\rho})\leq (1/L^{\rho})$. We also have $K, L >0$. Then we verify that the following inequality holds:
$$(1-a)^{b/\rho}(1/L^{b})\leq Q_b^{-1} \leq (1/L^{b}) $$
$$\rightarrow (1-a)^{b/\rho}(1/L^{b})\leq [a (1/K^{\rho}) +(1-a) (1/L^{\rho}) ]^{\frac{b}{\rho}} \leq (1/L^{b}) \tag{1}$$
by raising throughout to the $\rho/b$ power to get
$$(1-a)(1/L^{\rho}) \leq a (1/K^{\rho}) +(1-a) (1/L^{\rho}) \leq (1/L^{\rho}) \tag {2}$$
which indeed holds, obviously, given the assumptions. Then go back to the first row of $(1)$ and
$$\lim_{\rho\rightarrow \infty} (1-a)^{b/\rho}(1/L^{b}) =(1/L^{b})$$
which sandwiches the middle term in $(1)$ to $(1/L^{b})$ , so
$$\lim_{\rho\rightarrow \infty}Q_b = \frac {1}{1/L^b} = L^b = \big[\min\{K,L\}\big]^{b} \tag{3}$$
For more, see this answer.