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

how about 'Standard-bearer'?

A standard-bearer is a person (soldier or civilian) who bears an emblem called an ensign or standard, i.e. either a type of flag or an inflexible but mobile image, which is used (and often honoured) as a formal, visual symbol of a state, prince, military unit, etc. (Wikipedia)

Here's some reference to suggest the importance of a Standard bearer in battle.

The Standard Bearer should never surrender his Banner, if he, during a battle, could no longer safeguard the Banner; he should wrap his body into the Banner awaiting the final slash. (Safeguarding the 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.

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formality of the terms is appropriate to the formal, symbolic significance of the role, so I don't anticipate it being superseded by a modernization anytime soon.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)

Bannerman

From Wiktionary:

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

Colour guard

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

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