When Googling "science definition," the first thing that came up was "The intellectual and practical activity encompassing the systematic study of the structure and behavior of the physical and natural world through observation and experiment." After a little more research, I was surprised to see that this seems to be one of the stricter definitions of science (others may be as broad as "the state of knowing" or some such...), but it is one with which I can get on board. I tend to think of science itself in a very strict sense, as the process of developing and testing hypotheses. However, my big caveat is that there are many activities that are involved in (and are absolutely essential to) the practice of science that are not science per se according to that definition. This does not diminish their value to science. Some of this has to do with the acquisition of background knowledge that informs the hypotheses to be tested, while some of it is associated with making the results of inquiry available and comprehensible to the scientific community and the public.
So then is taxonomy art or science? With taxonomy, there is not a "right" answer, although there are plenty of wrong answers if one wishes to have a system that is informed by the results of scientific inquiry. Taxonomic units are all in some sense arbitrary. Although a group of organisms may form a "clade," whether we recognize that clade with a certain name is somewhat arbitrary. I personally like to think of taxonomic units being defined by specific innovations (morphological, molecular, ecological, etc.) that have changed the evolutionary trajectory of a group, but that rule is certainly not universally applied, and there could certainly be many alternative taxonomies even if such standards were applied.
For me, the argument for taxonomy as an art does not actually diminish taxonomy in any way as part of what we must do in order to be effective and responsible scientists. In fact, having this perspective on taxonomy can help to enhance the understanding of the significance of taxonomy for science. As scientists, we must use what we discover through the scientific process to help facilitate communication about natural phenomena. Taxonomy is a tool that we use to communicate ideas about organisms, so taxonomy is an absolutely necessary part of the pursuit of scientific truth, even if it is not "science" itself.
One test for me of whether taxonomy is itself a science in the very strictest sense of the word is whether it is directly involved in the process of hypothesis testing. One can use principles of phylogenetics, ecology, or molecular biology to test hypotheses, but taxonomic principles would not be used. When we begin to dissect some of the scientific questions that are often deemed "taxonomic questions," it can be argued that they are not actually taxonomic in nature, and that the taxonomic repercussions would really only be a byproduct of obtaining results through scientific inquiry. For instance, a question like "Is this a good genus?" is really asking something like "Do the species form a distinct clade?", which is a question that is evolutionary in nature. Likewise, the question "Do these individuals make up one species?" is perhaps just a way of saying "How can we properly apply a biological, morphological, chemical, ecological, and/or phylogenetic species concept to this group of individuals?", a question that draws on different fields of biology.
I can see that many systematists would hesitate to state that taxonomy is an art, because of what it implies. If it is an art, then it opens the door for people to say that people who do taxonomy are not really scientists at all. But a consummate scientist is not just someone who constantly tests hypotheses one after another without consideration for anything else. To be a scientist, one must also lay the groundwork for scientific pursuits, and defining the terms used to communicate ideas about specific units of the tree of life (whether or not it is itself an artistic pursuit) is crucial to the advancement of science.