Is a diversified portfolio with a beta of 2 twice as volatile as the market portfolio?

Yes. That's what it means. The "beta of 2" is a comparison to the market portfolio. The volatility measure is usually annualized standard deviation and the "market portfolio" is commonly the S&P 500 Index, but should be a broad index that is similar to the securities in the portfolio. The market portfolio used for a portfolio of international securities could be the MSCI EAFE Index, for example.