Manga

Just a heads up that you posted this question like an hour before the new weekly post went up, so it will probably be worth posting over there as well to get some fresh eyes on it.

I haven't read Natsume, but I have watched most of the anime (partway into season 7 right now). Some series I can compare it to based on that:

• Mushishi - Another series I watched the anime for instead of read the manga. It's similar in structure to Natsume in that it is a series of mostly self-contained stories and there are relatively loose threads that carry over between stories like recurring characters or questions about the MC's past. Instead of dealing with yokai, the MC deals with lifeforms called mushi, that other people can't see and cause problems for normal people unless they learn to coexist. Sound familiar? It can tend to be a bit darker in tone though and it doesn't always have a happy ending. If you go the anime route, I can vouch that it is excellent and one of my favorite anime ever.

• Star Tripper - This might seem like an odd one to recommend after Natsume due to the sci-fi setting, but it is pretty laid-back and chill like Natsume. Additionally, the MC is often getting involved with new people and learning about their lives in the process (like Natsume does). There is an undercurrent of uneasiness that belies the charming artstyle though, like not everything is alright... It's early still, but I have really liked it so far.

• Jeweler's Maid - Another one that might not seem similar at face value. However, this one is similar to the above recommendation in that it is an anthology series where the main character, the titular Maid Eliya, gets involved in people's lives and helps solve their troubles through the power of gemstones. It is a very chill series with some great art.

• Mieruko-chan - If you like Natsume for the yokai, then perhaps Mieruko is more up your alley. It is stuffed to the gills with spirits and creatures that only our MC, Miko, can see. It has a much more explicitly horror tone to it compared to Natsume, but the art is fantastically unsettling and the colorful cast is fun.