I think the main difference here is that your dog/cat don't have markets that they can be traded in relatively easily. There aren't pools of liquidity for which

I think the main difference here is that your dog/cat don't have markets that they can be traded in relatively easily. There aren't pools of liquidity for which they can be exchanged. Also, your dog and your friend's cat aren't like anything else. The example you constructed is superficially more similar to the NFT market anyway. If dogs and cats were fungible, and there was a market for fractions of both that could be exchanged in and out of, and contained hundreds of thousands of USD in standing liquidity, and the value of your dog was $1 million, and the cat was also $1 million, then sure. Swapping the dog for the cat is approximated by $2 million to the same accuracy that many assets (houses, stocks, etc) are described in notional value.