Covariance Is Meaningless for Unpaired Data

Today, through chatting with GPT and Gemini, I suddenly realized that the sample covariance
$$
s_{ij} := \frac{1}{n-1} \sum_{i=1}^n (s_i - \bar{s}_i) (s_j - \bar{s}_j)
$$
is meaningful if and only if $s_i$ and $s_j$ are paired.

If they are not paired,

any arbitrary pairing would lead to an arbitrary and meaningless covariance value. (from Gemini)

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For a real life example, imagine that you want to investigate the production of rice of two different farms. You got $n_1$ bags of rice from the first farm, and $n_2$ bags of rice from the second farm. Each bag has its weight. Here, $s_i$ are weights for bag of rice from the 1st farm, while $s_j$ are weights for bag of rice from the 2nd farm.

Obviously,

any arbitrary pairing would lead to an arbitrary and meaningless covariance value.