Carter's, Old Navy, H&M, Gap, and Target's Cat & Jack all overlap on these size-by-age ranges within an inch or two. The bigger variation is between cuts (regular, husky, slim) and between countries (US "12mo" runs slightly smaller than EU "74cm").
Buy by weight, not month label. A chunky 4-month-old often needs 6-9mo; a petite 9-month-old may still wear 3-6mo. The clothing tag's lower bound usually matters more than the upper.
Sizing source: brand size charts (Carter's, BabyGap, H&M); height ranges from WHO length-for-age 50th percentile.