Origin of 7 year sabbatical leave cycle
It seems to not be a coincidence. A glance at the Oxford English Dictionary shows that &quot;one-in-seven leave&quot; probably started at Harvard, but did not come with the name sabbatical.  If it makes you feel better, the rule was that this was the highest frequency allowed, and not all leave requests were expected to be granted if there were too many in one department at the same time.
Soon after Harvard 1880 policy, Wellesley in 1886 adopted a one- in-seven policy that called the year off a sabbatical year.
The name almost certainly is an allusion to Toric law relating to leaving land fallow one year in seven.  See Exodus 23:11 or look up Shmita.
The connection between revitalizing farm land and revitalizing the scholarly mind has been made clear by many.  Bertrand Russell wrote &quot;Every university teacher ought to have a sabbatical year (one in every seven) to be spent in foreign universities or in otherwise acquiring knowledge of what is being done abroad.&quot;  (On Education, especially in early childhood, 1926, by Bertrand Russell.)
The stricture of always going to Europe is not to be found in the US these days.  In modern times, I have heard of explicit rules that you have to physically go to another university. Of course that was in Antecovidian times.
In summary, the name refers to a one in seven cycle.  Just as a semester is not exactly half a year, the exact meaning of one in seven is subject to interpretation.