Variations have existed almost as long as trading cards have. The first notable variations were probably Old Judge photo variations, which date back to one of the first trading card sets! Many (most?) people, including me, consider those unique cards, though. However, jump ahead a couple decades, and the Big Tobacco card sets, such as T206, are well known for their different advertising backs.
These days, variations are a marketing gimmick, but back then, they were just circumstantial. And looking at Japan's card history, there are similar ... um ... circumstances.
Menko cards have come with a wide variety of variations. Back in the 1950s and 1960s, menko was a game and the fronts weren't so important. Different backs meant different games that could be played, which resulted in some card fronts being found with different style backs. And the opposite - recycling backs - meant that less time could be spent on designing the backs, and manufacturers could just churn out a new series of the latest popular stars with half of the effort. Similarly, I'm sure printing methods resulted in intentional mixing of backs and fronts in some series.
But beyond that, menko cards seem to frequently come in a variety of basic styles on a theme. Again, recycling concepts saved time at the drawing board. Take the round menko set JRM 1 and its five cousins.
One of the earliest pro baseball menko sets, JRM 1 was issued in 1948 and features 12 cards which are 2-11/16" in diameter. The tops are red, the bottoms are green. This set has a variation which indicates that it may have been reprinted. Specifically, there are no parentheses around the team name, and the line between the two background colors is yellow instead of black.
A little earlier than JRM 1 is JRM 1a, issued in 1947. The only real difference in design is the font, though these are a bit smaller. Again, this set has a variation in that it was issued in two different sizes (1-7/8" and 2-1/4" diameters).
Also in 1947 is JRM 1d, where the cards are larger than JRM 1a (but smaller than JRM 1), at 2-3/16" diameter.
Then, in 1950, a couple more sets with this design were issued. JRM 1b has two types, both which are the same size as JRM 1 (2-11/16" diameter). The first uses the same background style as JRM 1 (red over green), with black and white photos and more professional-looking text. Type II uses the same photos and text, but the backgrounds are solid blue, pink, or red - with each player having all three variations!
Finally, JRM 1c is a larger version of JRM 1b Type I, just larger (3-11/16" wide).
All of the sets are at least somewhat rare, but the 1950 sets are tougher to find, with 1c being the most difficult. In some cases, players are found in multiple sets, but the cards aren't all parallels of each other, excepting JRM 1b's Type I, and the three colors for Type II. And not all players in Type I appear in Type II, so it is a partial parallel.
I also have a JRM 1 and JRM 1a in my collection, so I have three of the six types so far. The quest goes on!