The short answer to your question, I think, is that the uniforms, standards, etc. of the Roman legions were red, the sacred color of the war god Mars. Since Rome is so closely associated in the popular imagination with warfare, the color red has come to be thought of as "Rome's color."

But the real significance of red - that is, of the various hues we collect under our culturally-specific definition of "red" - in Roman culture is harder to define.

It should be said, first, that the Romans themselves were aware of the vagueness of their terminology for the color red. A passage in the Attic Nights of Aulus Gellius (2.26) records a conversation between two very learned and excruciatingly pedantic men, in which the sophist Favorinus notes:

"For the color red (rufus) does in fact get its name from redness, but although fire is one kind of red, blood another, purple another, saffron another, and gold still another, yet the Latin tongue does not indicate these special varieties of red by separate and individual words, but includes them all under the one term rubor [redness], except in so far as it borrows names from the things themselves, and calls anything 'fiery,' 'flaming,' 'blood-red,' 'saffron', 'purple' and 'golden.'" And so on and on.

In short, though the Romans may have associated various pigments of red with widely disparate fields of reference, the poverty of their terminology leaves us a little colorblind.

A few things, however, seem clear.

Red was an important sacred color.

Beyond its associations with Mars, the color red had ancient and complex connections with the concepts of blood, heat, and summer (or so the Oxford Classical Dictionary informs me). Supposedly, the first two circus factions in early Rome - the Reds and the Whites - were symbolically linked with summer and winter. Or so the early Christian writer Tertullian claims in the middle of a vitriolic condemnation of chariot racing:

"The charioteers were naturally dressed, too, in the colors of idolatry; for at first these were only two, namely white and red: the former sacred to the winter with its glistening snows, the latter sacred to the summer with its ruddy sun: but afterwards, in the progress of luxury as well as of superstition, red was dedicated by some to Mars, and white by others to the Zephyrs..." (De Spectaculis, 9)

(Deeper hues of) red came to be associated with the symbolic associations of "purple"

Roman purple was really a deep (violet or scarlet) red (at least as we define the color). The great prestige of purple in the ancient Mediterranean derived from the difficulty of procuring it: no fewer than 12,000 sea-snails had to be crushed to produce 1.5 grams of the dye. Roman senators, famously, wore togas bordered with purple. Triumphators, and later emperors, wore a purple toga embroidered with gold.

Since the Romans had different words for purple and red - "purpur" and "rufus" - they obviously recognized a distinction between the deep red gained by crushing murex shellfish and the less vibrant reds produced by various other natural dyes. The associations of purple with power and prestige, however, probably bled over into the significance attached to other deep reds. Late antique sumptuary laws forbade private citizens to wear clothes resembling the emperor's (Cod. Theod. 10.20.18; 10.21.3) - a sure sign that some uppity citizens were doing just that.

Finally, red was the color of Mars, and thus of the Roman army.

From a very early period, Rome regarded Mars as the patron of the legions and their commanders. Generals and high-ranking officers wore the paludamentum, a cloak that was usually dyed either purple or red.

At least in the imperial era, soldiers seem to have been issued three tunics: one for everyday use (probably red or undyed), a white one for ceremonial occasions, and (at least in some periods) a red battle tunic (tunica russa militaris).

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So: the meanings of the color red in Roman culture were complex, and closely tied to the concepts of war and power. And since we tend to view Rome through those lenses, we're always seeing red.

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You know, there are so many funny coincidences in this world. I just finished reading an excellent article ("Le Role des Etiquettes de Plomb Dans le Travail du Textile a Siscia" - or, for those who don't do French, "The Role of Lead Tags in Textile Work in Siscia") about something quite similar to this topic overall. While these tags don't necessarily prove anything, and are a relatively small (sample size of ~1200) snapshot of Roman textile trade in the region (Siscia is/was a town across the Adriatic from Italy - in ancient Pannonia, the middle of modern Croatia). We're not entirely certain of the purpose of these lead tags - they were used to mark textile shipments with a weight and a "suggested retail price," along with a name (and they were often reused, which leads to some of them being etched into essential illegibility). The name bit is what we're unsure about, actually - whether the name is meant for: it could be the workshop/manufacturer, the owners, the individual merchant, or even the slaves/workers who were unloading the objects with the tags.

But those fun tidbits aren't the relevant part (THOUGH THEY ARE COOL). The tags very commonly are marked concerning red articles of clothing - and we're talking all types of clothing here: Lana, Pannum, Tunica, Sagum, Paenula, Palla, Palliolum, Lodix, Banata and Abolla (Essentially clothing of different styles for both men and women, including cloaks). The best part? The prices varied as much as the different styles! Let's look at some examples, eh?

Okay, so in the image above, you can see what they actually looked like. Don't mind the handwriting, that's Roman cursive and it's notoriously messy. The first tag we see says this:

DONATA
PANUM
IIMATINUM

and then on the back, that funky symbol that I dunno if I can recreate. That basically means 4 denarii. So this tag is for a pannum (or bunch of them - also, a pannum is a type of shirt/jersey worn by charioteers or those wanting to cheer for them. Or just those who wanted to wear a shirt), for Donata (again, not sure what the name is for), and the IIMATINUM is a fun word - the II is calligraphy shorthand for a long "ee" sound, and this is actually a Greek loanword that seems to have been smoothly incorporated into the Latin lexicon. Haematinum is essentially "blood red." So this blood red pannum was selling for 4 denarii each. Definitely not a cheap garment, depending on the date. We don't know the dates here, as these tags in particular were literally just dredged en masse out of a canal in the early 20th century.

Anyway, let's keep looking at some more! #2 in that picture:

AFRI
NUS SILIN
DI PAN
NUM

And on the back, you see the AIIMA (more shorthand for that aforementioned haematinum colour), but look! This pannum (same type of clothing!) only costs ONE denarius. Now that's a Wal-Mart brand if I've ever seen one. Maybe the previous one is Hudson's Bay? The front refers to an "Afrinus Silindi," which is DEFINITELY not a Roman name - the article assumes that he was a traveling merchant (literally "peregrine"). Next one gets EVEN MORE SHORTHANDY:

TVNI
CA IIM I P VII

...and then on the back, has the name (Cadida) and the price for each tunic (7 denarii - faaaancy). The front declares that these are tunics (tunica), IIM (shorthand for...you guessed it...haematinum) and 7 pounds (again, going to conjecture, but this is probably the weight of the package to which this was attached. Or maybe it's a measure of how heavily dyed - i.e. the shade - these tunics were. 7 denarii was EXPENSIVE. Anyway, you get the idea on these ones (the next couple are for tunica and pannum in that same shade). However, #6! #6 is different.

FIIRVGIN
PANVM

And then the back has the denarius mark plus an S - indicating a semi-denarius (2 sestertii). This one doesn't have the name on the front, but it DOES state the familiar panum. The colour, though - that one's different (yes I know the scribbling is bad, but trust me). This one's ferrugineum - iron, or rusty red. The name there is pretty self explanatory (more brownish). Next tag (#7) is another different sort!

PVRPV
RIIVM

and then the back:

F II R
SACVM

and then the funky little marks. I'll explain those first. So here, you can see the mark of the denarius, followed by a III. So it's 3 denarii, but then it's followed by an S and a few horizontal dashes. The S, as you remember, is a "part" symbol, while the little dashes are clarification as to the fraction. Three horizontals means three quarters, so it's 3 and 3/4 denarii. Now, about the fact that there are two different colours here. That's not super unusual - multiple clothes could be in the same bundle, or they could be from slightly different bundles. The price is right in the middle of Walmart and Hudson's Bay, so we can call this H&M clothing. The FIIR is that ferrugineum (rusty) colour, but the PVRPV is one that probably caught your eye! While that is indeed short for purpureus (purple), it's certainly not that fancy Tyrian purple. More likely, it's a knockoff shade of deep, purplish red. Think knockoff luxury clothes that you can find today (H&M). And then the garment in question is just "sagum" - so a cloak.

Anyway, I've probably bored the pants off of whomever read this far - sorry, I got carried away. This stuff is super cool! Some of the other tags refer to a "peppery" colour (assume grayish?), and one even to a cerulean (Blue's my favourite colour). But there are as many shades of red in this selection as all of the other colours combined, which, while I can't paint the whole Roman empire with a broad brush and say that this clearly shows that all Romans loved red, does indicate that it was popular, at least in this part of the world.

Lemme know if you (or anyone else!) want to chatter more about Romans dyeing! (hehe)

EDIT: Also, basically the only extant Roman shield that we have is very, very red. It's literally a sample size of one (which is to say, highly unhelpful), but it's what we've got.