Appointed to a chair vs to the chair

Solution 1:

The American Heritage Dictionary (https://www.ahdictionary.com/word/search.html?q=chair) notes, in an early definition, that a chair is a position of authority.

In American universities and similarly in European ones, there are heads or chairs (chairmen or chairwomen) who are appointed to lead the department.

There are also endowed professorships that are called chairs, of which several may exist in one department. Those named after a famous person or a donor are usually preceded by “the,” e.g., the Arthur Burns Chair in Economics.” Endowing a position is a common way to make a donation to a particular department of a university.

In the case of an endowed, named chair, the position is unlikely to have any authority beyond the professorship.

Solution 2:

In the case of Peter Singer, the term chair refers to an endowed chair, which is a term synonymous with endowed professorship. Note that if, in the sentence you are asking about, you use endowed professorship instead of chair, the indefinite article is not at all strange:

Less violent protests took place at Princeton University when I was appointed to an endowed professorship of bioethics.

Therefore, the indefinite article is not strange in the original sentence, either.

Discussion

First of all, what is an endowed chair? Wikipedia has a good definition:

An endowed professorship (or endowed chair) is a position permanently paid for with the revenue from an endowment fund specifically set up for that purpose. Typically, the position is designated to be in a certain department. The donor might be allowed to name the position. Endowed professorships aid the university by providing a faculty member who does not have to be paid entirely out of the operating budget, allowing the university to either reduce its student-to-faculty ratio, a statistic used for college rankings and other institutional evaluations or direct money that would otherwise have been spent on salaries toward other university needs. In addition, holding such a professorship is considered to be an honour in the academic world, and the university can use them to reward its best faculty or to recruit top professors from other institutions.

See also the related question on the Academia StackExchange, here.

Next, here we read the following:

Princeton University appointed [Singer] the first Ira W. DeCamp Professor of Bioethics at the University’s Center for Human Values. Philanthropic giving to Princeton paid for Dr. Singer’s transformation. The Center for Human Values was established with money donated by Lawrence Rockefeller, and the Ira W. DeCamp Foundation endowed the chair bearing its name.

And here it is stated that

The choice of Peter Singer, a world renowned animal rights advocate, for a chair in bioethics at Princeton University's Center for Human Values, has agitated the genteel college town as well as the field of bioethics.