To me it means you have common interests with the other person and like each other also. It can also mean that person makes you happy or is very interesting to you. Someone you like talking to or being around as you learn from them or just have a good time together. You enjoy them. Good close friends enjoy each others company.
Enjoying someone's company simply means that you being with someone or spending time with them.
The word company has several different meanings, but in this context, company refers to being in the presence of a second person.
There are several things that might make you enjoy someone's company:
From an etiquette point of view, being in someone's company also has some social "rules" which it is considered polite to follow. There are certain things that you might not do or say if you were in someone's company, which would be perfectly fine if you were alone.
For example, walking around the house in our underpants eating Coco Pops.
Different types of "company" also demands different levels of formality. If you're in the company of someone important, like your boss or a member of the royal family, you might adopt a different demeanour than if you were in the company of friends down in the pub.
Didn't find the answer you were looking for?