The answer is yes, and it has a long history. The 'flag' is more properly understood as a standard, and came in many forms. It is depicted in your two paintings as a national flag, but even more important were regimental regalia that allowed generals and units to organize effectively. The history of having a standard bearer in your army goes back thousands of years and stems from that necessity. Would a standard bearer be actively fighting/throwing themselves into danger as in the paintings? Well, that is of course dramatic license to a degree, but if your regiment is tasked with taking a particular objective, the standard bearer's prominent flag being on that objective is a signal to the general and the army that you have succeeded.

As to when these practices were discontinued? Basically the same time that set-piece battles battles did. Standard bearers are useful precisely because they allow such battles to be effectively run by a general, but as army sizes increased and communication technology advanced, the concept of the standard bearer shifted to a ceremonial role, not an active military one.

Carrying the Colours was regarded as an honour (just as the loss of the Colours - a regiment having two - was a disgrace, and a great feat for the capturing side). cf Sergent Ewart at Waterloo amongst others

In the British army (continuing today), each regiment has two colours, the Queens (or Kings) and the Regimental. The Regimental colours are usually of a design related to the history of the regiment, while the Royal colours generally feature the Union Flag and the Royal cypher.

Typically, the colours would be carried by an Ensign (the junior officer rank) and a guard provided by senior NCOs with pikes to protect them.

Digging around a bit, the last occasion I could come up with for a British regiment taking their colours into battle was at Laing's Nek during the First Boer War in 1881.