TikTok might offer a way for you to repost your favourite videos on the platform, with a new experiment with a "Repost" button. I've spotted it on my personal TikTok account. And as social media consultant Matt Navarra noted, these reposted videos are only going to show up to your followers and not on your profile. The option is also just available on the For You page and not on your Following page.
So far, I can't find a way to access them once I've refreshed my feed. But when you've reposted a video, it'll show above the username that you've reposted the video (as you see in the screenshots above). I've tried reaccessing the video through Hank Green's profile, but the indicator that I've reposted isn't visible there.
The feature is still in beta, so that might still change around or work differently in the future. As 9to5Mac noted, the test shows that the company is aware people are looking for ways to share their favourite TikTok videos on the platform. There is no way to do so right now outside this feature.