Learn English – the connection between motherhood and apple pie

Stéphane's suggestion, “to mingle with the crowd”, may well reflect the intended meaning of the speaker of the phrase. But note that in English, the phrase “take a bath” often is associated with a stock-market loss (1, 2); since the European idiom is not well-known in the U.S., I'd expect American English speakers to either associate your phrase with taking a loss in the market (along with a crowd of other investors) or to interpret the phrase literally.

The earliest citation I can find is Marion Harland's Alone (1856), but I don't suppose it's an "origin".

Free, white, and twenty-one" sang Emma, cheerily. "Twenty-one! In four years, I shall be a spinster of a quarter of a century!"

The fact that it was well-established long before OP's 1930s movies is attested by this sentence in the Transactions of the Annual Meeting from the South Carolina Bar Association, 1886

And to-day, “free white and twenty-one,” that slang phrase, is no longer broad enough to include the voters in this country.

There were still black slaves in some states in the mid 1800s, so obviously being free and white was a meaningful part of "I can do what I want and no one can stop me". But unless it refers to the "freedom" to vote, I don't know what the significance of reaching 21 would have been at the time.