using namespaces in headers is bad style. at least move this https://github.com/bjoekeldude/Mini-Log/blob/fba2a2b813fc0a29cd53c90cd2905fa4c6bfd51c/miniLog.hpp#L10 into your miniLog namespace

https://github.com/bjoekeldude/Mini-Log/blob/master/miniLog.hpp#L15 your already in the miniLog namespace, maybe just name it msgType? https://github.com/bjoekeldude/Mini-Log/blob/master/miniLog.hpp#L16 also maybe just message?
https://github.com/bjoekeldude/Mini-Log/blob/master/miniLog.hpp#L17 already in a struct called message maybe just _time, _type and _str? also a special naming convention for member variables is a good idea, underscore prefix is not a bad choice.

https://github.com/bjoekeldude/Mini-Log/blob/master/miniLog.hpp#L21, since you are making a copy of the input string any way, would be better to take by value and move it into the member variable.

https://github.com/bjoekeldude/Mini-Log/blob/master/miniLog.hpp#L32 strftime returns the new length, would be good to resize the string to the proper length.

s.resize(std::strftime(&s[0], s.size(), "%Y-%m-%d %H:%M:%S", std::localtime(&now)));

https://github.com/bjoekeldude/Mini-Log/blob/master/miniLog.hpp#L36
maybe make both of these into getTimeString functions into member functions of the message

https://github.com/bjoekeldude/Mini-Log/blob/master/miniLog.hpp#L59 should probably not be copyable, what would copying a logger even mean? use = delete after a constructor to remove it.

https://github.com/bjoekeldude/Mini-Log/blob/master/miniLog.hpp#L60 path should probably be private

https://github.com/bjoekeldude/Mini-Log/blob/master/miniLog.hpp#L62 since your storing a copy might as well take by value and move it in