Nah they're a single molecule. While they do have a mechanism to "reproduce", they cannot react to stimuli of any kind, or evolve. Of the 7 commonly accepted traits of life, viruses have 5-6 depending on where you stand with them not being able to reproduce on their own. (In comparison, while a tapeworm or other parasite might need a host, they bring their reproductive equipment with them).
Prions have 1 of those traits. They can't regulate an internal environment as they cannot have one, they lack any kind of organizational trait, they have no metabolism (the other one viruses lack), they do not grow, they don't adapt to their environment, and they do not respond to stimuli.
A digital thermometer has organization and responds to stimuli, so it's more alive than a prion.
The paper indicates the forms are specifically limited, in mice there were 15 specific forms they could take.
But still, they evolve between the forms, so yeah, they are equally alive as a digital thermometer. Now they just need to get their act together to beat a tamagotchi.
Evolution came up with sexual reproduction as a mechanism to keep the gene pool varied and varying. In this sense the female and male sexual organs constitute one single reproductive apparatus.
Another ingenious mechanism is the bee/flower or fly/flower or bat/flower, duos of completely separate species in beautiful, poetic symbiosis. Birds with seeds is another one.
The bees (or flies or bats) reproduce on their own, but the flower has turned those species into part of its' own reproductive organs, in a way.