Here is the problem, Euclidean distance is not recommended for datasets with many zeroes (like matrices of species/site), as there is the risk of the abundance paradox (Orloci, 1978). Whereas to calculate environmental distance (i.e., using Temperature and Precipitation variables) the Euclidean distance is widely used. The problem is these are not easily comparable. Is it correct to use Hellinger distance on environmental variables (normal distribution)?
It is pretty easy to compare two dissimilarity matrices (assuming that is what you mean by compare?).
For example, you could ordinate the dissimilarity matrices separately and compare them with Procrustes rotation. Or there is the method of co-intertia analysis which extracts axes that maximise the covariance between the two data sets (cf PCA which extracts axes of maximal variance in the one data set) subject to axes being orthogonal. Co-inertia is based on Euclidean distances so you could apply the Hellinger transformation to the species data and leave the environmental data untransformed, or you might transform some of the env data using say a log transformation.
Mantel's (partial) test can also be used to compare associations between two or more dissimilarity matrices.
What isn't comparable? If you mean because they are measured on different scales, this is why people use correlation based PCA; so you could standardize/center your variables and then calculate euclidean distances. Keep in mind that the so called "Hellinger transformation" is simply the square root of relative abundances; does it make sense to do that to your environmental data?
Is it possible to do this, yes. Does it make sense, I don't think so.
There are clustering algorithms that won't work with other distances (e.g. k-means), but there are also a lot of clustering algorithms that can work with arbitrary distance functions. If you don't have thousands of instances, classic "hierarchical clustering" will work.
If your data set is larger. DBSCAN and OPTICS may be a good choice.
You may want to have a look at ELKI, which is very flexible when it comes to distance measures. They have a tutorial on how to add a custom distance function, in case they don't have Hellinger distance yet.