Yes, sorta.
The British system used in WW2 was organized into Regiments identified by a name, such as Wiltshire Regiment. These regiments had several numbered battalions that served in different units and theaters. A British regiment could have as many as 12 battalions, including training battalions and territorial battalions. So the entire regiment did not serve in one place. These battalions were identified such as "2/Wiltshire Regiment".
A brigade was organized by several battalions or regiments. A brigade was usually identified by a patch that was worn on the shoulder. For example, the Highland Division Brigade [blue rectangular patch with "HD" in a circle], was comprised of battalions of Scottish regiments.
Example of a brigade composed of battalions of same regiment.
169th Infantry Brigade - Gen. L. O. Lyne
2/5 West Surrey
2/6 West Surrey
2/7 West Surrey
(Where 2/6th West Surrey designated the Second battalion of the Territorial Army.)
The US Army at the beginning of WW2, had two brigades in an infantry division with two regiments in each brigade. Then they reorganized their divisions into "triangle divisions" and reduced the size to elimnate 1 regiment and the brigades.