Yes. Ancient coins were struck in the following way. The minter takes a specific weight of a specific metal or mix of metals in a coin shape. However, it's not so easy to make blank coins of consistent weight by hand- so, most coins in a single coin type will be of slightly different weights and sizes. There won't be major differences if the mint is doing it's job right, but the differences will exist.
Now, to actually mint the coin, the mint will need two dies. Dies are metallic implements with a circular end. Think cone shaped, as far as we can tell, though only forged dies survive from the ancient world since real dies tended to be destroyed due to fears of forgery. On the circular end, an engraver will have engraved the design of either the obverse (in much ancient and modern coinage, usually the side with a face on it) or reverse of the coin. The second die will have the engraving of the other side of the coin. So say we're talking about this coin of Tranquillina from the 3rd century A.D.:
One die would have the obverse (the profile of Tranquillina) and the other would have the reverse (concordia and the marriage) engraved on it. A hot blank coin would then be placed on one of the dies. The mint at this point takes the other die and hammers it into the coin, and because the coin is hot it absorbs both engravings at once. However, the mint now has to do this for many, many coins. It's not easy work. So, the minting might get sloppy; the die might be hammered into the coin off center, meaning that part of the engraving will be missing from the coin. At the same time, the die that is being hammered into the blank coin tends to deteriorate at a faster rate than the die on which the coin rests. So, for single issue, you might have one hundred coins with the one obverse die and one reverse die, and then another fifty with the same obverse die but a different reverse die, and on and on it will go (those numbers aren't at all an accurate estimate of how long a die would last, by the way). And each die is likely to be slightly different, even with the same design, since engraving the exact same thing image isn't easy. Thankfully, this allows brave souls called numismatists to do die studies, which can allow us to trace the relative dating of a coinage issue by examining the changing dies.
I hope that answers your questions about why so many ancient coins look so weird! Someone with knowledge of modern minting will have to explain how our coins are minted with much more precision.
Source: Grierson, Philip. 1975. Numismatics. Oxford: Oxford UP.
