I'm not a lawyer or a legal scholar, so I'm going to ignore the exact legal prescriptions and instead explore the spirit of the law. How much are moderators like employees?
The tasks that moderators do are tasks that moderators could be paid for: handling complaints (typically submitted via flags), handling support requests (typically submitted via meta), performing tasks that others cannot do after verifying that the action is warranted (editing, closing, reopening, deleting, …), providing guidance and leadership to the community (again typically via meta), etc. Community managers do those tasks as part of their paid job.
Moderators never have any obligation to work on a particular schedule. Moderators do not have any obligation to work, withone significant exception: moderators on Stack Overflow are expected to spend a minimum of time. I'm not aware that this has ever been enforced, but it's possible that some SO mods who stood down voluntarily because they couldn't keep up with the expected level of activity would have remained if there hadn't been any pressure for them to maintain a high level of activity. Even with this caveat, in terms of obligation to perform, moderators are different from employees, who have to work when the employer requires it.
Moderation is more of a gig. The legal status of paid contractors is a point of legal and political debate in many jurisdictions. Regardless of whether habitual contractors should be treated like employees, they always expect to be paid for completed work.
So what moderators do can be considered work. Do moderators work as directed by the company? The moderator agreement — which has no legal force, since it isn't a contract (the most that Stack Exchange can do if they don't like what a moderator did is dismiss the moderator, and they legally can do that even if the moderator hasn't violated the agreement) — states:
I will abide by the then-current Terms of Service of $SITENAME, and other moderator policies made available to me,
(…) I acknowledge and agree that I am an independent volunteer moderator to $SITENAME and I am not an employee, agent or representative of Stack Exchange Inc., and I have no authority to bind Stack Exchange Inc. in any manner.
These statements do not fully characterize moderatorship on Stack Exchange. Moderators are not only expected to abide by the terms of service and other policies, but also to enforce certain policies, such as the code of conduct. Moderators do not have authority to speak on behalf of the company, but they are often perceived to do so. Moderators are go-betweens, representing the company to the community and representing the community to the company. While moderators have considerable latitude in how they apply directives from the company, the company sets the boundaries. For example:
As a moderator, you're held to a higher standard and are expected to set a positive example for your community. (…) After that, if you cannot in good conscience follow it, reach out so we can find a replacement for you.
There are situations in which you cannot be a moderator and do nothing. Specific performance is expected. The only punishment is dismissal — but if an employee fails to perform as directed, the punishment is rarely worse than dismissal.
I can't speak for everyone, but for my part a major reason why I resigned¹ was because I was in some respects an agent of the company. This doesn't by itself imply a subordination relationship: I would have reacted in the same way if I'd been a member of a non-profit organization that had done something unconscionable and I'd lost any hope or interest of changing the organization from the inside.
This brings us to another criterion: are moderators purely subordinates of the company, or are they to some extent participants in decision making? Here the answer is very clear: moderators don't have a say. The company decides.
Finally, who benefits from the work of moderators? That's a bit hard to delimit. There are benefits to the world in general, to the community in particular and benefits to the company. The world benefits because the repository of knowledge is better curated. The community benefits because moderators wield some of the tools that make it work and serve a social function to regulate the community. The company benefits because moderators are effectively delegates of the company: the company could hire enough moderators to cover the work that moderators do, but doesn't thanks to the volunteer moderators.
I think I can speak for pretty much all current and former moderators in saying that the reason we do or did this is to benefit the world and the community, not the company.
So are moderators similar to employees? To some extent. Certainly enough that there is no clear-cut yes or no answer. There may be a clear-cut answer within the scope of a specific law, but I'm looking beyond that.
Now I never expected to be paid and I would not have been a moderator for pay. I'm sure that the former is true for all of us and the latter is true for most.
Paid moderators would have a very different relationship with the community. Even though moderators are sometimes perceived as agents of the company, they are first and foremost community members. A monetary relationship would destroy this. Having moderators who are and remain from the community is important for community cohesion and for the health of the community. People who face the same problems that other community members are facing are more likely to take decisions that benefit the community.
Moderators should be giggers, but giggers on behalf of their community, which is itself a volunteer endeavor. The current status is that moderators are giggers on behalf of both their community and the company. This creates ambiguity which is detrimental to everyone: to the company (who may have legal and financial obligations that it hadn't bargained for), to the moderators (who face conflicting responsibilities when the interests of the company diverge from the interests of the community), and to the community (because moderators cannot fully act on their behalf).
This needs to change. The governance of Stack Exchange needs to change. We need to distinguish the platform (the servers, the software), managed by a for-profit company, from the communities (the content, the moderation), managed by a non-profit. And then moderators would be volunteering for a non-profit organization, which carries no expectation of pay.
¹ Background: I was a moderator for over 8 years, across 5 different low- to moderate-activity sites (1 to 3 at a time).