Let (X,d,μ) be a non-homogeneous metric measure space. In this setting, the author proves that the bilinear Calderón-Zygmund operator T˜ is bounded from the product of weighted Morrey spaces Lω1p1,κ(μ)×Lω2p2,κ(μ) into weighted weak Morrey spaces WLνω→p,κ(μ), and it is also bounded from the product of generalized weighted Morrey spaces Lω1p1,φ1(μ)×Lω2p2,φ2(μ) into generalized weighted weak Morrey spaces WLνω→p,φ(μ), where ω→=(ω1,ω2), ϱ∈[1,∞), ω→∈AP→ϱ(μ), P→=(p1,p2) satisfying 1p=1p1+1p2 for 1≤p1, p2<∞. Furthermore, via the sharp maximal estimate for the commutator T˜b1,b2 which is generated by b1, b2∈RBMO˜(μ) and T˜, the author shows that T˜b1,b2 is bounded from the product of spaces Lω1p1,κ(μ)×Lω2p2,κ(μ) into spaces WLνω→p,κ(μ), and it is also bounded from the product of spaces Lω1p1,φ1(μ)×Lω2p2,φ2(μ) into spaces WLνω→p,φ(μ).