Multiple Scrum Teams working on the same product do not have to have the same Sprint start date. The Scrum Guide does not prescribe any synchronization of Sprints across multiple teams1. However, it is recommended that the teams align their Sprint cadence and duration, and coordinate their dependencies and integration through a shared Product Backlog, a common Definition of Done, and frequent communication2.
This way, the teams can deliver a coherent and valuable product Increment every Sprint[3][3].
References:
1: The Scrum Guide, November 2020, p. 9
2: Managing Products with Agility, Scrum.org, accessed on December 16, 2023
[3][3]: Understanding and Applying the Scrum Framework, Scrum.org, accessed on December 16, 2023