While I can't necessarily answer this for medieval warfare, I can give the purpose of a flag (usually called a 'colour' in contemporary sources) in gunpowder warfare: it was a visual marker on which troops could organise themselves.

Every battalion would have at least one colour specific to them, often with unit-specific symbols and a list of that unit's battle honours that were carried into battle (the British army had a 'regimental colour' and a 'king's/queen's colour', the latter being the union flag with the regimental number embroidered on it). This served as a rally-point for soldiers, letting them see where the centre of the battalion's front rank should be. It allowed troops to reform after crossing broken ground, after a melee, or simply know where they were supposed to be on a smoke-filled battlefield.

This is the origin of the tradition of 'trooping the colour', now most famously practiced by the battalions of the British Household Division: the troops needed to see their colour to be able to identify it in the heat of battle and use it as a marker.

Since a colour was a symbol of the unit and its honour, it also served an important moral function: Napoleon famously granted Eagles to his regiments to carry their colours, often presenting them personally. This bound the army to his regime, and the capture of an eagle was a great dishonour to a French regiment and therefore a coveted achievement for France's enemies. Indeed, many surviving Napoleonic eagles are actually held by British regiments, as they wanted to preserve their prizes, as opposed to the restored monarchy, which wanted to destroy the symbols of the old regime.

As for making you an easier target: it absolutely did. Being part of the 'colour party' was a highly dangerous job, and the junior officers who bore the colours (at least in the British Army) had a very high attrition rate, especially in major battles like Waterloo.

Sources:

Redcoat: The British Soldier in the Age of Horse and Musket by Richard Holmes

Swords around a Throne: Napoleon's Grande Armée by John R. Elting