I am miscalculating the Asymptotic Bode Plot
I got what was I missing thanks to everyone who commented. I was not taking the pole at the origin into account. However, the magnitude of this pole starts with a slope of -20dB/decade at infinity, and at w = 0.1 rad/s, it is at 20 dB magnitude. Adding that 20 dB to the 15 dB I had already calculated, yields the starting magnitude as 35 dB, just like on MATLAB. Big thanks to everyone who helped!