This is an important point. In countries that adhere to the Vienna Convention, mandatory signs are an important category of signs that correspond to certain traffic rules. 
is part of this convention.
By contrast, in the United States, mandatory signs officially don’t exist at all, and neither does the legal basis for them. The closest thing is the occasional “must” on a regulatory sign, such as 
obligating cyclists to make a two-stage turn. But regulatory signs about access either say that you may or you may not, so there isn’t an obvious opportunity to use access values such as designated and use_sidepath.
Some jurisdictions have antiquated laws requiring cyclists to use a sidewalk if provided, but people don’t tend to map such laws because they’re uniform across the whole region and not well known. Or the laws are full of obscure conditions about age, age of the accompanying adult, etc. that no one knows how to tag.
I tend to use bicycle=designated instead of bicycle=yes because iD’s Cycle & Foot Path preset uses bicycle=designated foot=designated. If that’s imprecise, then at least there’s little practical harm compared to omitting foot=* from a highway=cycleway, which does break routing.
That said, there are situations where I find it important to distinguish between bicycle=designated and a mere bicycle=yes, for example:
I would interpret the icons on this sign as foot=designated horse=designated bicycle=designated wheelchair=designated moped=no hunting=no. Although it’s perfectly legal to bike on the parallel residential road (bicycle=yes), you would “go out of your way” in doing so. (The ⅓ route shield calls for a route=bicycle relation but doesn’t affect the access tags in any way.)
This trail is officially a multi-use path for pedestrians and cyclists. An occasional park rules sign reminds pedestrians to keep off the paved trail and stay on the sandy shoulder when possible. This rule isn’t strong enough to relegate pedestrians to foot=no shoulder:foot=designated, but it does establish cyclists as having the right of way. (Unusually, cyclists even have the right of way over motor vehicles at crossings.)
This expressway=yes has a bike lane. Can you see it? It’s indistinguishable from a shoulder, except for an occasional “Begin Turn Lane – Yield to Bikes” sign at an off-ramp. The local highway department has a policy of keeping the bike lane as low-key as possible. They don’t want casual cyclists to find out about it, because the heavy traffic at 50 mph would be hazardous. Nevertheless, this is a popular commuting route for more experienced cyclists.
In the West, some stretches of Interstate freeway allow cyclists because there’s no other connection, short of dismounting and climbing a mountain. But even though these freeways have all the relevant signs allowing cycling, they would be only bicycle=yes because they only allow cyclists by necessity.
Aside from these cases, if you prefer to tag a bike trail as highway=path instead of highway=cycleway, then bicycle=designated is the most obvious way to refine it as a bike trail as opposed to a trail that happens to allow cyclists. Iterative refinement is also possible with path=*, but that key is about physical construction rather than usage.
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