The summary does a poor job of explaining Amazon's point (which isn't to say it's a good point anyway). Amazon doesn't seem to be saying that Alexa has rights, so much as it's saying that you or I or anyone else who owns an Echo has rights, and that those rights would be trampled if this warrant was served.
Their line of argumentation seems to be the following:
1) People have a First Amendment right to say and express anything they want in the privacy of their home (which is true)
2) If people aren't secure in their privacy, we've stripped them of their right to express themselves freely (also true)
3) If the police could hear anything anyone has said, it would have a "chilling effect" because people wouldn't be secure in their privacy (yup)
4) The police are asking for days' worth of audio without any direct evidence it has anything to do with the crime (true, I guess)
5) Thus, if they granted the police access to those recordings, they would be compromising the rights of Alexa users everywhere (wait...what?)
The problem with their logic is, of course, that the police aren't forcing anyone to buy an Alexa device. If I choose to purchase a device that, by design, records everything I say, then I've voluntarily sacrificed my right to privacy in exchange for the benefits afforded by the device. It's not the police's fault that I've done so, and they're entirely within their rights to seek a warrant for the information that I've served up on a platter.
This isn't blanket police surveillance, like Amazon appears to be asserting. This is a blanket devaluation of and disregard for the importance of privacy. Amazon is trying to protect us from the consequences of our poor choices, not because they're interested in protecting our interests, but rather because their business depends on having no consequences for using their products. If people actually understood just how creepy Alexa and similar products are, they'd stop inviting them into their homes. Amazon is worried that a case like this will shine light on Alexa's privacy-destroying behavior.
Sacrificing one's privacy should never be treated lightly.