According to Jeff Bezos, it's a hardware button, and various sources seem to agree from the teardowns

A forum post at the EEVblog forums quotes a video featuring Jeff Bezos, the founder of Amazon:

In this video about Jeff Bezos being interviewed by Walter Isaacson at around the 6 min mark, Bezos claims the mute button on the Amazon Echo is physically connected to the mic amplification circuit, making it impossible to enable again via software.

This is also supported by a reddit thread in which it is said that "Basically it is a physical analog connection that cuts off circuit flow to the mic." Another commenter added:

No voltage to mics when mute is on. You're correct as well about the state of mute being software controllable. That said, the state of the LED under the button is tied electrically to if the mics are on (same circuit), so there's no possible way the mics can be powered without you knowing it.

That said, those sources aren't particularly clear on which models they're referring to. Taking a further look at the teardown linked in the forum post may be interesting to verify this.

Another source that supports the 'hardware button' theory is the Apple Insider site, which discusses a previous Echo vulnerability. It notes that:

Despite gaining access to the "always-on" microphone, the hack cannot get around the physical mute button on the device, which disables the microphone completely. This switch is a hardware mechanism that cannot be altered with software, though it is feasible that with extra work this button could be physically disabled by a determined attacker.

Disappointingly, the iFixit teardown doesn't include a good image or any commentary on the mute button circuitry for the proper Echo device. Even so, there's a substantial amount of evidence that it may be a hardware button after all.