Property testing in other metrics?
Yes, there is!  I will give three examples:


Given a set S and a "multiplication table" over S x S, consider the problem of determining if the input describes an abelian group or whether it is far from one.  Friedl, Ivanyos, and Santha in STOC '05 showed that there is a property tester with query complexity polylog(|S|) when the distance measure is with respect to the edit distance of multiplication tables which allows addition and deletion of rows and columns from the multiplication table.  The same problem was also considered in the Hamming distance model by Ergun, Kannan, Kumar, Rubinfeld and Viswanathan (JCSS '00) where they showed query complexity of O~(|S|^{3/2}).
There is a large amount of work done on testing graph properties where the graphs are represented using adjacency lists and there is a bound on the degree of each vertex.  In this case, the distance model is not exactly Hamming distance but rather how many edges can be added or deleted while preserving the degree bound.
In the closely related study of testing properties of distributions, various notions of distance between distributions have been studied.  In this model, the input is a probability distribution over some set and the algorithm gets access to it by sampling from the set according to the unknown distribution.  The algorithm is then required to determine if the distribution satisfies some property or is "far" from it.  Various notions of distance have been studied here, such as L_1, L_2, earthmover.  Probability distributions over infinite domains have also been studied here (Adamaszek-Czumaj-Sohler, SODA '10).