"Period" means many things. One meaning is related to "phrase," but "phrase" also has more than one sense, not necessarily to do with breathing. From Dolmetsch online:
Period: a complete musical thought, concluded by a cadence, having from two phrases, each usually two to eight bars (measures) in length, called the 'antecedent' (often ending in a half cadence) and the 'consequent' (often ending in an authentic cadence)
Phrase: a short musical idea similar to a sentence in spoken language; also a style of performance that gives shape to the musical phrases
Notice the lack of any reference to breath. Players of wind and brass interments, and singers, must consider breath in their phrasing, but this isn't necessarily the same sense of "phrase" that a composer is concerned with. The phrases that make up a period are structural phrases, theoretical or analytical in nature, not necessarily related to breathing.