Prince of Persia: Rival Swords

Based on the search results, your description is partially accurate but contains some claims that cannot be verified.

Confirmed information:

Prince of Persia: Rival Swords is indeed a port of Prince of Persia: The Two Thrones for PSP and Nintendo Wii, released April 3-5, 2007[1] The Wii version utilized motion controls, with combat mapped to Wii Remote and nunchuck motions rather than button inputs[2] The content is largely identical to The Two Thrones with only minor differences[4] The game was rated T for Blood and Violence (compared to the original's M rating), indicating content was modified[1]

Unverified claims: The search results do not confirm the specific bonus content you mentioned for the PSP version. While one user on GameFAQs noted seeing gameplay of "the prince fighting enemies on what looked like a beached ship" that they didn't remember from The Two Thrones PC version, suggesting possible extra content exists[3], the search results provide no details about bonus levels, chariot races, or a multiplayer racing mode for the PSP version.