What do we call a person in a war who holds the army's flag?

An old-fashioned/specialist¹ term for this is standard-bearer. It is now more widely used as an idiom for someone who represents a cause of any sort, but the original meaning was the person who carried an army's standard or flag. From Oxford Dictionaries:

1. A soldier who is responsible for carrying the distinctive flag of a unit, regiment, or army.
   1.1 A leading figure in a cause or movement.

This is from the now-rare meaning of standard

4. A military or ceremonial flag carried on a pole or hoisted on a rope.

I don't know how widely-understood the literal meaning of the term is, but military history buffs would definitely know it.

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¹ Old-fashioned/specialist in the sense that the component words are used in ways that are no longer standard outside of specific contexts. Standard, in particular, is not used in everyday language for flag, and even bearer is used most commonly in more formal and ceremonial contexts, as in a pallbearer or a ring bearer, but not a mail bearer (mail carrier is far and away more common; see, for example, this Ngram).

That said, the term is neither obsolete nor archaic; in the proper context, it is the correct, still-current term. And the formality of the terms is appropriate to the formal, symbolic significance of the role, so I don't anticipate it being superseded by a modernization anytime soon.

In the British Army this person would be a Colour Sergeant or Staff Sergeant ("the colours" being an alternate name for the regimental flag)

While it's probably a bit esoteric for your purpose, the rank of ensign has historically had this duty - although the rank seems to have been long since abolished in most militaries.

(According to the linked Wikipedia article the rank is also used in Star Trek.)

Bannerman

From Wiktionary:

A person assigned to carry the colours, standards or guidons of a military unit.

Guidon bearer

It is an honor to be the guidon carrier for a unit, known as a "guidon bearer" or "guide". He or she stands in front of the unit alongside of the commander (or the commander's representative), and is the rallying point for troops to fall into formation when the order is given. In drill and ceremonies, the guidon and commander are always in front of the formation.

https://en.wikipedia.org/wiki/Guidon_(United_States)

Colour guard

Often there is a single bearer, but also guards to protect the bearer.

https://en.wikipedia.org/wiki/Colour_guard

I'm surprised no-one has mentioned flag bearer yet. A google image search for this term gives images more reliably in keeping with what the OP wants than Standard bearer does. The latter term is several times more popular on ngrams, but, as the images reveal, it frequently refers to a Roman style standard, which was not made of cloth, but instead was a solid emblem at the top of a pole.

Flag carrier also finds images close to the OP's meaning, but it also finds images related to national airlines, because this is another meaning of this phrase.

If you really want a single word definition, then how about "signifer".

A standard bearer of the Roman legions

A bit archaic I know!

Color–bearer

One who carries a color or standard especially in a military parade or drill