It makes sense that each attribute should be a taxonomy – eg roots, leaves, seeds – but you don’t need a separate taxonomy for months. Instead, the months should be the terms of these taxonomies.
The second argument to register_taxonomy in each case should be the name of your custom post type. This will mean you’ll get a box for each taxonomy on the ‘Edit Plant’ screen. And in each box, a list of months (you’ll need to populate this yourself).
Obviously this makes it straightforward to get the months of interest with respect to a certain plant, but it’s perfectly possible to go the other way. For instance, if you wanted to get all of the plants which were of interest in a given month, you could construct this query quite easily.