I am viewing a risk report of a hedge fund and the portfolio vega seems to be a plain summation of the vegas of the different asset classes the fund invests in (i.e. Equity, Credit etc)

As far as I know, vega is additive when referring to the same underlying with a similar maturity, therefore, the assumption of the hedge fund is too simplifying?
Is there any kind of 'mathematical' proof regarding why this is wrong?

And the main question, what is the 'correct' way to do this?