In the seventeenth century, England absolutely had traditions of both mockworthy names and of mocking them. For example, an actual Jonson's (that would be Ben) play "Bartholomew Fair" had some fun:
Quarlous: His Christian name is Zeal-of-the-land?
Littlewit: Yes, sir; Zeal-of-the-land Busy.
Winwife: How! What a name's there!
Littlewit: Oh, they all have such names, sir: he was witness for Win here--they will not be called godfathers--and named her Win- the-fight: you thought her name had been Winnifred, did you not ?
Winwife: I did indeed.
Littlewit. He would have thought himself a stark reprobate if it had.
Elsewhere, Jonson lays waste to biblical buzzword names like Tribulation, Restraint, and Long-patience.
In the early and post-Restoration 17th century, unwieldly biblical and Bible-ish phrases as names bore the brunt of jokes. Most of this was driven by anti-puritan/nonconformist sentiment or the good politics of being anti-puritan in public, since "Job-rakt-by-the-ashes" and similar names were strongly associated with puritans. But they weren't exclusive to it, as Sir Faithful Fortescue (and his son Faithful and his nephew Faithful...) shows. The point is, the focal point of ostracizing and ridiculing religious noncomformity provided a cultural shorthand for mockable names.
Even some of those sympathetic to puritans--as David Como argues, we need to talk about a spectrum rather than a distinct sect even before the English Civil War--seem to have viewed the names as a shade of ridiculousness, as minister Thomas Adams did in 1629:
it is a petulant absurdity to give [children] ridiculous names, the very rehearsing whereof causeth laughter. There be certain affectate names which mistaken zeal chooseth for honour, but the event discovers a proud singularity....as if they would tie the goodness of the person to the signification of the name. But still a man is what he is, not what he is called; he were the same, with or without that title or that name. And we have known Williams and Richards, names not found in sacred story, but familiar to our country, prove as gracious saints as any Safe-deliverance, Fight-the-good-fight-of-faith or such like, which have been rather descriptions than names.
In contrast, Jo(h)n Jo(h)nson is a name that pops up frequently in the Calendar of State Papers volumes released by the British PRO for the 17th century. Just as a sample, the 1637 volume mentions two distinct John Johnsons; the 1629 one also mentions two. They seem to come from a variety of social levels. One of our 1637 mentions is a new arrival to London. A 1629 representative is a university fellow.
Although John Johnson ("spelled variously", says the index) appears to have been the most common double, I did find some Richard Richardsons as well.
All of this is to say: mere incidence of the name--even though given the limited sample, the number of John Johnson is probably significant--isn't enough to say people didn't do a double-take at the pseudonym. After all, the pattern of puritan names being mocked had roots in people's actual, baptismal-record names, too.
However, if Fawkes had intended to be ridiculous, he had a whole barn of name options to choose from that would have been instantly and acceptably recognized as such in popular culture. Instead, he picked a name with a reasonable amount of currency in London at different levels of society.
Further Reading: