Mostly, yes, but there are several differences. Kokuji and shinjitai are only used in Japanese. Of the roughly 35000 hanzhi, only a few thousand are used as kanji. Simplified Chinese characters are only used in mainland China. And the pronunciation is different.
Just want to add a point, due to the opening up and rise of china especially the People's Republic of China (PRC), as opposed to Republic of China (ROC) Taiwan. Simplified chinese has been introduced outside of Mainland China. Politics is messy here, so I'll just share some parts of history that I understand (but may be contentious) and stay away from the political talk.
Many overseas Chinese education have shifted towards Simplified Chinese in their education for political alignment of ASEAN countries and China in part with the aim of Asean Free Trade Area as the influence in trade and culture is strong due to geographical proximity. A good example is in South-East Asian overseas chinese population (natives as of 2nd generation of immigrant descent onwards), Singapore and Malaysia adopted Simplified Chinese since 1969+.
As for the main question, I'll say the glyphs in the radical and parts that make up the kanji characters are mostly of Chinese origin. As ポールおじちゃん mentioned, there are also japanese-created kanji such as Kokuji and Shinjitai. The writing system of both countries/cultures (Japan and China) continued to evolve when china and japan closed up their borders to foreign trade in some periods of history.
Nowadays with Mainland China's cultural revolution and introduction of simplified chinese characters, there's less similarities with Mainland China's Simplified Chinese Hanzi and Japanese Kanji, compared to Taiwan's Traditional Chinese Hanzi and Japanese Kanji.
There has always been tension between the writing systems of PRC and ROC.
One famous example is the Hanzi/Kanji for Love:
爱 vs 愛
Look above, hopefully your OS can read both simplified chinese and traditional chinese characters, the left-side character is simplified chinese (PRC), right-side character is traditional chinese (ROC) and kanji (JP).
The ROC actually used this criticism slogan in their traditional chinese writings system campaign:
"How can love exist without the heart (心)?"
