My understanding is that the principal quality of a prion is that it behaves like a prion in cells- the logic is a little circular.
A few of these criteria for acting like a prion (in yeast, where prions are most common):
- Reproduces itself by "templating" (refolding of non-prion versions of itself into prion versions)
- Sensitive to protein denaturing agents (which get rid of the prion protein conformation).
- Dependence on protein chaperones which reproduce prions by releasing prion oligomers from larger prion aggregates. These oligomers go on to "seed" new prion aggregates.
- Forms insoluble protein aggregates consisting of many copies of the prion protein.
An important subtlety here is that proteins with identical amino acid sequence can be both prion and non-prion in the same cell. It is a biochemical consequence of a specific protein structure, rather than simply a feature of the protein sequence. Thus, a prion domain confers the ability to become a prion, not necessarily "prion-ness" itself. This Scientific American article might be helpful regarding this point.
As an example of how these criteria are applied, this paper uses some biochemical and genetic methods to screen for the presence of new prions in wild yeast.
As a review of how prions oligomerize and form their classic aggregates, this paper might be helpful.
It is certainly true that creating fusion proteins with candidate prion domains is a common way of confirming that those domains are competent to become prions (see e.g. this paper). But there is usually a higher standard of evidence to show that some protein actually is a prion- usually by demonstrating the existence of heritable protein aggregates mentioned in point (4).
Hope that helps.