True screen printing emerged much later in China but other types of textile printing would have been well known to Rome. Resist dyeing, stenciling, and hand-painting on fabric would have been done as well.

However, I've never heard of men wearing a printed insignia like a t-shirt as part of a military uniform (assuming that's what is pictured here?). A lot of historical fiction shows prioritize the overall look and feel of the design over historical accuracy. If the designer wants the modern audience to think a man is rough and sexy it is easier to do that by putting them in something similar to the modern equivalent (ex. a muscle shirt) than to put them in a period accurate tunic and hope the audience doesn't just see a man in a dress. It's the same reason you see more men in Medieval shows wearing pants rather than leggings. The designers aren't bad at research, they're making a designerly choice.

That particular pattern looks to me like it's been block-dyed. A block is carved into a sort of "stamp" then painted and pressed onto the fabric. It's a beautiful technique that's still in use today (my bedspread is from Balochistan and is block-printed and I love it).