It has been possible to do partitioning in PostgreSQL for quite a while — splitting what is logically one large table into smaller physical tables. Since version 10, a huge leap was made with the introduction of declarative partitioning, and more improvements have come every year since.
Sharding is a different story — splitting what is logically one large database into smaller physical databases. The primary tool for this in the PostgreSQL ecosystem is the Citus extension. But you can also handle the sharding logic at the application level, as recent posts from the likes of Notion and Figma have described. Somewhat confusingly, some forms of sharding are sometimes referred to as vertical partitioning, including by the team at Figma.
Read on for a few thoughts on when to perform each action and what the costs and benefits are.