RNA polymerase Sigma subunit: transcription factor, coenzyme, or what?
No.
I don’t think anybody would call the sigma factor of bacterial RNA polymerase — or indeed any other protein — a cofactor or a coenzyme, the subset of cofactors that are organic (the original poster is wrong in his declaration about cofactors, or at least one reputable text book disagrees with him).
I could justify this statement by quoting a definition from the section linked above:

“The catalytic activity of many enzymes depends on the presence of small molecules termed cofactors, although the precise role varies with the cofactor and the enzyme.”

by saying proteins like the sigma factor are not small.
However it is more interesting to me, and more useful to the naive reader, to make some general remarks about definitions in biochemistry and molecular biology.
What are definitions good for?
Definitions of terms are useful for scientific discussion or argument to enable those involved to be sure they are talking about the same thing. They often arise at the stage of a field of science when things are crystallizing after a period of uncertainty or lack of clarity. Even so they may lack precision. Cofactor just means “something else that is needed for certain enzymes to work” — biology is not physics.
Definitions are also useful for students and their teachers. They allow students to try to build a rational picture of a subject that is new to them. They allow teachers to systematize a subject (or lazily set rote-learning type MCQs instead of asking students to explain things).
The student new to the subject might find it helpful to consult glossaries such as those in Molecular Cell Biology, Molecular Biology of the Cell and Pearson’s Biological Glossary.
What are the limits of definitions?
The problem in a fast-growing field like modern molecular biology is that the period of resolution — when old men can form committees to give names to things — seldom lasts. Biology is anarchic, and new things come along that don’t fit into the neat pigeon holes that have been so carefully assembled. Nor do the anarchists or revolutionaries care. They find new things, describe them, and then get on with the next experiment. Why spend one’s time reading yesterday’s newspaper?
But the problem for the student comes when he discovers that everything does not fit into the neat categories he’s been taught, that there is a world beyond lactate dehydrogenase and metal proteases.
Sigma factors — what’s the problem in categorizing them?
The “classic enzymes” studied in early biochemistry catalysed simple chemical reactions between small molecules. However catalysis also occurs in more complex polymerization processes like nucleic acid and protein synthesis, that have the complexity of repetition along a polymeric substrate, and the necessity of having a specific start and stop along that substrate. It turns out to do that, that rather than enzymes such as RNA polymerase being a single catalytic protein they are complexed with other proteins that are necessary for certain of these functions. In the case of proteins like the sigma factor(s), this is complicated by the fact that these accessory proteins do not remain permanently associated with the catalytic protein subunit.
The poster, in his title, asks whether the sigma factor is a transcription factor. The answer to this is yes… and no. And calling it a transcription factor doesn’t really help much. Yes, it is a transcription factor because it is necessary for transcription and different sigma factors can impart a certain specificity to bacterial transcription. No, because the term transcription factor is mainly used for the much more specific proteins in eukaryotes that work in a rather different way.
What to call it? A subunit of the enzyme, is sufficient for Berg et al.. An accessory protein might perhaps add something. But don‘t worry too much about naming it, learn how it works.
Postscript: Other fruitless terminological disputes you may wish to avoid

Why is NAD regarded as a coenzyme and ATP a co-substrate?
Why is it epidermal growth factor, not epidermal growth hormone?
Why is it called vitamin D when it’s actually a hormone?
Should the term, molecule, be applied to double-stranded DNA or the individual chains?
How long must an oligopeptide be before it is called a polypeptide or protein?

No doubt there are others that I have missed that can be found in the list archives.