I'd be interested in any time period and other countries as well. I'd assume it would be more common in the earlier part of the war since this would be when the army was constructing whole units, as opposed to replacing losses in already existing ones.
During the early parts of the First World War recruitment for the army was still based on the past regimental recruitment style; a man wishing to serve in the army would join by signing up at a Regimental Depot or Recruiting Office. That means the man was able to choose which regiment he served in. Obviously, men would join the closest regiment to their homes i.e. being from East Lancashire you'd probably want to join the East Lancashire Regiment. So if a man and his friends/siblings all decided to join the army they'd want to serve together, therefore they'd join the same regiment.
Later in the war, the British had a severe recruitment problem. So few men were enlisting that the government passed the Military Service Act of 1916. The act conscripted all single able-bodied men between 19 and 41 years old, which was later adjusted to include married men and lowered the minimum age to 18. Conscripts were not allow to choose which regiment to join, however their preference for the army or navy was taken into account.