My two cents:
Let's agree that a derivative is long an underlying if the payoff of the derivative increases with the price of the underlying $S$.
Then buying a variance swap is going long the volatility of $S$, and it is also long the variance of $S$. It is a convex payoff of the volatility of $S$ and linear in the variance of $S$.
Similarly, buying a volatility swap is also going long the volatility of $S$ and going long the variance of $S$. A volatility swap's payoff is concave in variance of $S$ and linear in volatility of $S$.
That's all, nothing about vol of vol.
An option on a varswap or volswap or realised volatility or realised variance would be long 'vol of vol' at any time before expiry, where just as there are different notions of vol there would be different definitions of 'vol of vol'.
An instrument that would be long expected / forward start risk-neutral 'vol of vol' before and at maturity is trading the spread between the VIX future and the forward starting volswap of the same tenors. Because at maturity of the VIX future the payoff is the difference between the VIX (square root of variance swap) and the spot start volswap, which is greater than zero by Jensen's inequality.
EDIT
Following dm63's and Newquant's answers below, for which +1 for both, an edit to try to settle this good question from the OP.
Let $\bar\sigma$ be the annualised volatility over the interval $[0,T]$. Then
$$
X_t := E_t [ \bar\sigma^2 ]
$$
is the varswap price, and
$$
Y_t := E_t \sqrt{\bar\sigma^2}
$$
is the volswap price.
Now suppose that we take the varswap as the base instrument. Then relative to the varswap the volswap has vol of vol (to be precise to the vol of the varswap) exposure since
$$
Y_t = E_t \sqrt{X_T^2}
$$
However, the varswap, as base instrument, does not have exposure to the vol of vol, just like the SPX spot price has no vega.
If we take the volswap as the base instrument, then the volswap has no vol of vol exposure, but the varswap does (to the vol of the volswap), since
$$
X_t = E_t [Y_T^2]
$$
Regardless of the base/reference instrument , the difference between the two has vol of vol exposure.