The short answer to your question is an emphatic "yes". This was a change in perspective, as for ~50 years in the middle of the 20th century science (and philosophy) tended to avoid questions about consciousness.
However, the last two decades of the 20th century saw an increasing interest both among philosophers and scientists in understanding ourselves, and in particular our consciousness. And this project has continued to grow in the 21st century. A key advocate for this effort was Francis Crick, who set out to inspire and recruit a generation of neurologists to try to crack this last mystery. He outlined his own views in The Astonishing Hypothesis: http://www.consciousentities.com/crick.htm
Crick, and many of his inspirees, ASSUME neuro reductionism. And therefore treat "scientific study" as == "neurological characterization". The author of your Guardian piece is a materialist reductionist as well.
However, the achievemetns of the neuro-reductionist project have not been nearly as successful as Crick and his initial inspirees hoped. One of the reasons is a logic problem, with the Guardian article mentions only obiquely.
- IF nero reductionism is true
- THEN all causation attributed to consciousness is fully
explainable in purely neurological terms
- THEREFORE the apparent causation for "conscious" events is basically
illusory
- AND THEN consciousness is an illusion or an epiphenomenon
But the author (and most other people who intimately experience it) is not willing to accept consciousness is an epiphenomenon. However, as he is not willing to give up on reductionism, he explicitly declares a limited goal of exploring mechanisms of consciousness, and has given up on understanding why it exists at all.
A significant number of consciousness researchers have abandoned neuro reductionism. Christof Koch, Crick's own primary collaborator is among them. He is now an advocate of Integrated Information Theory, which is an architectural and organizational approach to consciousness, as opposed to a neuro-reductionist one. IIT, Global Workspace Theory, and Higher Order Processing, among others, are IT inspired approaches to investigating consciousness, that equate consciousness to running a particualr algorithm, rather than having a particular set of neurons. This is Algortithmic Identity Theory, rather than Neural Identity Theory. Koch is also willing to consider either pan-psychism, or property emergence, rather than material reductionism, as alternative ontologies.
Another major area of science thinking in investigating consciousness is to revive idealism. Here is a recent collection of thinking about experimental data relative to consciousness, and how it supports an idealist ontology: https://www.bernardokastrup.com/2015/03/review-of-beyond-physicalism.html
So, while "YES -- science can study consciousness", your linked author's approach is already looking to be at best useful but not relevatory, and there are at least two other major scientific paradigms being deployed to engage in this investigation.