Is Fuji and Fujitsu the same company?

No, both are Japanese companies, but are separate.

Fujitsu Limited (富士通æªå¼ä¼šç¤¾ FujitsÅ« Kabushiki-Kaisha?) is a Japanese multinational information technology equipment and services company headquartered in Tokyo, Japan.[4] It is the world's third-largest IT services provider measured by revenues

Fujifilm Holdings Corporation, commonly known as Fujifilm, (富士フイルãƒæªå¼ä¼šç¤¾ Fujifuirumu Kabushiki-kaisha?) is a Japanese multinational Photography and imaging company headquartered in Tokyo, Japan.