I'm speculating because I don't know what legal reasons would be at play here, but my assumption is that it is either tax related, or that the primary means for proving employee status are tax records. If that assumption holds, the IRS policy of rounding to the nearest dollar would necessitate a pay that rounds to $1 instead of $0.

That makes you wonder if they could then just pay $0.51 so that it rounds to $1. Even if that were the case, my guess is that a nice even dollar is just easier.

A $1 salary in the context of minimum wage/minimum salary laws is certainly legal for business owners:

Under a special rule for business owners, an employee who owns at least a bona fide 20-percent equity interest in the enterprise in which employed, regardless of the type of business organization (e.g., corporation, partnership, or other), and who is actively engaged in its management, is considered a bona fide exempt executive.

That means a business owner can be an exempt employee (no minimum wage/overtime requirement) without meeting the other criteria (minimum salary) that apply to exempt employees.

This Business Insider article from last year highlights a number of CEO's who are/were receiving salary of $1 or less. On the list are several that don't own 20% of their companies, which supports the notion that there are other exceptions to the minimum salary rule, but based on the DOL sheet linked above I don't see what specifically would apply to non-owners.

I like Hart CO's answer about rounding and tax implications. But I'm gonna come at this from another direction.

Even one dollar per year might not be legal

A case can be made to challenge the legality of the one-dollar-per-year salary. The US Department of Labor (DOL) sets a minimum salary for exempt employees. Beginning on 01 January 2020, this minimum is $23,660 annually. In this context, "exempt" primarily means exempt from receiving overtime pay.

So if the employee in question is exempt, they would not be able to receive a salary of one dollar per year without violating that DOL rule. On the other hand, if they are non-exempt, they would be subject to the more well-known minimum wage -- currently $7.25 per hour in the US, and higher in some jurisdictions. Plus, they'd be subject to hour tracking and overtime. (I suspect most of these are senior executives, and they probably work well over 40 hours per week.) Most wage and hour regulations round to the nearest 15-minute increment, and have a minimum number of paid hours per shift. Even if the minimum is fifteen minutes, that's $1.81 even if the employee in question only worked 15 minutes in the entire year.

Per the Wikipedia article linked in the question, many recent or current situations of this phenomenon are government employees. Perhaps the DOL regulations don't apply to government employees -- I haven't found anything on that yet.

The special case of the U.S. President

In the specific case of the president, the U.S. Constitution sets forth that the president is to receive a fixed salary, and then 3 U.S. Code § 102 currently has it set at $400,000 per year. That's why President Trump couldn't opt for zero (or one dollar per year) and instead is donating his salary to various organizations.