Was red actually a meaningful colour for the Romans? Did they dress their soldiers in red clothes like in movies and reenactments? Even maps constantly present the Roman territories in red. Is there a historical basis for this focus on red?
Hi! There's always room for more information, but FYI, there was a similar question recently - see here for previous responses
I went down the rabbit hole and couldn't find this referenced in the link provided, so I thought I'd throw it out there.
Regarding your second question, the answer is sometimes. Graham Sumner did extensive research into the nature of Roman military clothing and published his findings as (surprise, surprise) Roman Military Dress in 2009. As part of his research, he delved into the subject of clothing color, something which had been a hot topic in reenactment circles following the publication of an article by Nick Fuentes (It's been a long time since I've seen it, so I don't have a link....) which suggested that the color of Roman military tunics was exclusively white. Sumner examined a huge variety of sources in order to ascertain the true nature of tunic color in the Roman army and came to the perhaps disappointing conclusion (if you're looking for a straight answer) that there was no right answer. In summary, he found that a wide variety of colors were depicted in artwork, surviving documents, etc. as being worn by the Roman army at various points in its existence. Specifically, he found that, in his study, white was the most common, followed by red, with other colors (green and blue, for example) being fairly uncommon but present. Both his summary of colors present and the evidence he cited can be found in chapters 4 and 5 respectively in his book. From a practical standpoint, white makes a lot of sense as it's easier to launder (you can just boil it and not worry about color loss) and cheaper to manufacture (white wool doesn't need to be dyed), meaning that you can replace it more easily if it becomes soiled.
As far as why we associate the color red with the Roman legions, Sumner cites Isiodore of Seville as stating that the "old" army (he was writing in the 7th century) was always dressed in red [my paraphrasing] (page 115 of Sumner's book). That may be the root of the association, but I'm not sure. Interestingly enough, Isiodore also states that the Spartans dyed their tunics red to conceal blood, so that same passage may be the root of that myth as well.
As to why the empire is shown in maps as red (I presume you mean modern maps), as a part-time cartographer, I'll hazard a guess that it's just the bias of the cartographer. We tend to associated empires with violent conquest, so red is a good choice. Also, red sticks out like a sore thumb. Throw in the fact that red is associated with Rome in modern pop-culture, and you can have your pick of reasons. I wouldn't suggest reading too much into map colors unless you want a good post-modernist headache.