Oh, this is actually really fun question and I'm glad you asked it! I currently specialise in Roman numeracy, so what follows is my own speculation on the topic.

Firstly, let's note that, as you said, Roman society wasn't organised around decimals as much as our own. When it comes to measures and coinage, Romans (like the Greeks) actually used - as a rule - 12-based fractions and divisions most. It's not quite clear where does this practice come from - Romans had in their close Italic neighbours "more decimal" peoples, like Oscans, who used 10-based metric systems. But, the 12-based metrical universe has the benefit that 12 is divisible with 2, 3, 4, and 6, whereas 10 with only 2 and 5 - so a 12-based systems of thinking arguably has more practical flexibility. Also, there are c. 12 lunar months in a year, and all ancient calendars (at least around the Mediterranean) had 12 months, so 12 might have felt as rather natural number to define physical and cosmological realities of the world. I'm generally a bit vary of assigning any specially meaningful or "magical" connotations to any numbers specifically, because you can find them for almost all lower numbers in Roman culture, and certainly we can see also the number 12 being significant in various religious, historical and cultural moments (12 Olympians, 12 Tables of first laws, 12 or 24 books in epic poems, 12 lictors with 12 fasces to magistrates etc etc.......). The 12-based thinking can be easily seen in Roman Republican coinage divisions, and measures of volume, area, weight and length in use throughout the Roman period.

So, why is a 'hundred' also meaningful? This probably, at its very roots, has something to do with the rather universal human tendency to use the number 10 as a numerical anchor. I'm gonna believe Stephen Chrisomalis' (2010) Numerical Notation: A Comparative History and state with fair confidence that all known numerical systems anywhere around the world have been decimal. They might in some rare cases have been based around duplicates of ten, such as the famous sexagesimal (60-based) system used in sophisticated ancient Mesopotamian mathematics, but it still has a sub-base of 10 and, you know, the 10 is there. The simplest speculative answer to why exactly this is would be the very universal human experience of 10 fingers and 10 toes - finger-counting being the most accessible form of numeracy anywhere and in some cultures small numbers and fingers share names (I just recently read Gary Urton's fun book The Social Life of Numbers on Quechua numeracy and fingers loom large there). An ancient thinker, writing sometime in antiquity, known as Pseudo-Aristostle also noted that “all men, both barbarians and Greek, count up to ten” and (alongside more far-fetched theories) muses that this might be so because everyone has ten fingers (Problemata 15.3, 910 b23–911 a4). Also Romans used a 10-based numeral system, with a sub-base of 5: I, V, X, L, C etc...

So, in general, Roman numerical universe was a mixture of decimal and duodecimal (or in more Latin term, uncial) definitions. So coinage denominations, in relation to each other, could always be a mixture of 10-based and 12-based figures (is it all very confusing? Yes! A Roman called Volusius Maecianus wrote a whole pamphlet to Marcus Aerelius to navigate the metric madness of Roman Empire). Note that Augustus didn't do any overhaul to make the coinage system be purely decimal/100-based like most modern coinages tend to be, if you look at the equivalences, and also the weight standards continued to be duodecimal. This mixture of having to navigate 12-base and decimal thinking becomes clear in one passage of Horace, where he describes (I assume) a typical exchange between a teacher and a student:

Roman boys learn in long calculations how to divide as into a hundred parts. "Suppose Albinus's son says: if one-twelfth is taken from five-twelfths, what is left? You might have answered by now." "One-third." "Well done. You'll be able to manage your money. Now add a twelfth; what happens?" "One-half." Ars poetica (325-40).

The reason school-boys learned to divide by hundred was because this was common way of calculating and expressing interest; e.g. if you can divide by 300, you can calculate an annual Roman interest, in modern terms, of 4%. The second bit shows that school boys memorised by heard some sort of conversion tables (they didn't have handy numerical representations like 5/12, 6/12 -> 1/2 for fractions like this!) between fractions in a 12-based system.

Anyway, point is, both decimal and duodecimal ways of numerical thinking were always been there in Roman culture. Is there something special about hundred, then? Well, since 100 is 10 x 10 - the most intuitive and basic numerical anchor multiplied by itself - it becomes rather organically a significant numerical threshold. I don't know if there is any better reason to explain the existence of e.g. centuria, the basic unit of Roman military; or that the original Senate was thought to consist of 100 senators; or many other ways the number was significant in Roman society. There was also the association that 100 years was a sort of maximum lifespan of a human - related to this are the Saecular Games, where a saeculum of 100 years was understood to be a sort of fixed, cyclical era-period observed during Roman history. In imperial image-making, religiously observing the start of every new cycle with games became important for ushering a "new era". (In reality, lots about the religious history of Saecular Games was blatantly made up for propagandist purposes and emperors also manipulated the chronologies of the saeculum, e.g. Augustus defining it as 110 years and Claudius as 100). Combined to this was that emperor's would make every 100th anniversaries of the "founding of Rome" significant and extravagant celebrations.