The interpretation and units problem, ie the lack of an easily intuitive answer, is precisely why quants/econometricians etc. tend to shy away from talking too much about covariances [even if they are absolutely necessary; and frequently used]. Thus if anything involving covariances has to interpreted, let alone explained, the default is usually to express it in terms of correlation, which does have intuitive units: bounded [-1,1] with 0 = independence, etc.
Cor(1,2) = Cov(1,2) / ( sd(1) * sd(2) )
Cov(1,2) = Cor(1,2) * sd(1) * sd(2)
So the "units" here is a product blend of three measures, each with their own units: two volatilities and a bounded measure of association. As such, they exist but lack an intuitive explanation.
The closest one can do is to express the covariance as a marginal change in portfolio variance per unit change in the product of Weights 1 & 2. Which remains inelegant in the extreme, to be polite ;-)
Recall also that the traditional OLS beta can be expressed as:
Beta(1|2) = Cov(1,2) / Var(2) = E(d1) / d2
E(d1) = Cov(1,2) * d2/Var(2)
So a change of +1 in Asset2 has a +0.1 divided by its variance effect on Asset1. Which is the same as saying that a +1 sigma move in Asset2 has a 0.1 divided by its standard deviation on Asset1. Which is the same as saying (where Z=1 is a 1 sigma shock):
d1/d2 = Cov(1,2) / Var(2)
d1/z2 = Cov(1,2) / SD(2)
z1/z2 = Cov(1,2) / (SD(1) * SD(2)) = Cor(1,2)!
So the way to make the kind of statement you try to make above intuitive remains to translate your covariances into (intuitive) unitless correlations. A one sigma move in either 1 or 2 will have a marginal Cor(1,2) sigma effect on the other.
However you approach this, you always need to process the covariance via an additional metric (with its own units, whether absolute returns, vol-adjusted returns, or weights) to generate any intuitive explanatory outcome here. The traditional w.Cov.w formulation is efficient for predicting portfolio risk; but when it comes to interpretation and explanation, it fails big time. Which is why publications inevitably show the associated correlation matrices in preference. The two will always give you the same outputs/forecasts; with the choice between the two ultimately a question of prediction vs interpretation (ie presentational in nature).