It's now a somewhat archaic expression, but during the period of "Bringing Up Baby" (and before) it referred to a generous, unselfish act. In this case, David thinks Mr. Peabody is being noble for wanting to donate $1 million to the museum.
More about the expression here -
https://en.wikipedia.org/wiki/Play_the_white_man
I could read that today on Twitter.
shareLanguage does evolve, for better or worse.
Several years ago, I was working with some very young people (early to mid 20s), and one guy told me how a former supervisor really let him have it. I said, "oh, he really gave you the third degree, huh?" He looked puzzled and replied, "third degree what? You mean burn?" Another time, a young woman told me something incredulous. Shocked, I asked, "are you putting me on?" She stared at me and said, "putting you on...what?"
Welcome one and all to the 21st century....third decade in.