The members of the US cabinet are called secretaries. Why it is so?
In what other countries besides the US and pre-WWII USSR ministers were called by a different word (in the USSR they were called commissars)?
Because the US took its nomenclature from Britain, as "Relaxed" and subsequent commenters have said above. As to why Britain has the tradition of calling ministers and certain permanent civil servants by titles involving the word "secretary", according to this answer on Quora,
"Secretary literally means 'one who keeps a secret', reflecting the confidential nature of the role. The use of the word to mean 'office assistant' is far, far, more recent than the use of the word to mean 'senior official'. To this extent, a 'Company secretary' in the UK is a senior figure, either a Director, or of equal authority to a Director, and with a bunch of legal responsibilities for reporting / compliance."
The word secretary had a long life before it came to mean someone who sits in an outer-office and services a manager or group of managers with modern "secretarial skills". Prior to that it meant someone "skilled in letter writing".But it goes back long before that.
It derives from Latin secretarius, One who is entrusted with private or secret matters; a confidant; one privy to a secret. Obs. (Oxford English Dictionary)
The first example in the OED is from 1387. But in the 16th century one important sense was:
†c. Applied to those entrusted with the secrets or commands of God, or of a god. Obs.
1599 in R. Hakluyt Princ. Navigations (new ed.) II. i. 209 The mercifull God..commaunded his secretarie Abraham to build him an house in Mecca.
1647 N. Bacon Hist. Disc. Govt. 1 Their Priests, whom they [the Britons] accounted the onely Secretaries that God had on earth.
1657 P. Heylyn Ecclesia Vindicata ii. iii. §14. 164 There was no order and command of Moses, or of any other of Gods Secretaries.
a1727 I. Newton Chronol. Anc. Kingdoms Amended (1728) ii. 210
Thoth, the secretary of Osiris.
transf. 1641 Milton Reason Church-govt. 41 But were it the meanest under-service, if God by his Secretary conscience injoyn it, it were sad for me if I should draw back.
From there it was a clear route to being secretary to a monarch or ruler.
As I commented earlier, it's the case in Britain too, which probably inspired the US practice.
Adding another example, in France, while senior ministers are called “ministre”, junior ministers are called “secrétaire d'État”. Such a “secretary” has a smaller area of responsibility and is placed under a full minister but it's still a cabinet-level position, held by a politician. Similarly, in Belgium and in the Netherlands, a “staatssecretaris” is a kind of junior minister. It's also the case in Spain I believe.
In some German-speaking countries (Germany, Switzerland), a Staatssekretär is a top-level civil servant, something like a chief of staff. So it's not quite a minister but still a very prestigious job. The word “secretary” and its cognates also have many other uses referring to senior roles of various kinds, not only in politics.
The term secretary has been unfairly maligned, misused, and changed since the cabinet was first instituted. In the 18th Century almost all jobs were performed by men. A secretary was a prestigious position that conferred the trust, authority, and confidence of the person the secretary represented.
Lawyers were often the most likely place where a secretary could get their start. In the 18th century a letter could take months to arrive and was costly so sending a letter was not done the way we send off emails like they are free and worthless today. So while today we see typing up a email or even a letter and sending it off as nothing, there was a time that a secretary would carefully craft and write with calligraphic penmanship a letter that was considered a token of esteem by the recipient. Technology has made this task seem less important, and thus the position has been demoted in the minds of many. Though the legal secretary still exists, the businessman today can much more easily handle the tasks themselves.
As a result of technology a new role that used to be the other responsibility of the business secretary has arisen, the Project Manager. It used to be that a business secretary would handle the process of organizing of functions, and staffing of the necessary positions to support them. This role often conveyed the authority of the boss. This was also at a time when org charts were more linear rather than branching like a tree. The secretary often had the authority to hire or contract as needed for a job. Everything went through the secretary because the secretary's job was to manage these tasks and only those decisions that actually needed the boss's approval went up to him(they were all him's then). Today this position is likely titled director or Vice President at a modern day corporate level.
But between the time when the modern day corporate organization structure became popular, and the advent of the technology that made it easier to manage your correspondence yourself than having someone else do it, the secretary position was still needed just at a level we call the Admin Assistant today. So while the business, legal, and executive secretary still exist, their role today is not what it was in the 18th and 19th centuries.
But there is one place where the position is still one of importance, that is the political realm. Communication and authority are still very important when dealing both with the bureaucracy, and with other states. In this way the secretary of X for a state is still one of respect, and prestige.