This is the definition of superficial from OALD:
not concerned with anything serious or important and lacking any depth of understanding or feeling
What does concerned with mean here? There are four senses of concern from OALD I find that all seems to be appropriate to the context:
1 [often passive] concern somebody/something to affect somebody/something; to involve somebody/something
– Don't interfere in what doesn't concern you.
– The loss was a tragedy for all concerned (= all those affected by it).
– Where our children's education is concerned, no compromise is acceptable.
– The individuals concerned have some explaining to do.
– To whom it may concern … (= used for example, at the beginning of a public notice or of a job reference about somebody's character and ability)
– Everyone who was directly concerned in (= had some responsibility for) the incident has now resigned.
– Please pay attention because this information concerns all of you..2 concern something (also be concerned with something) to be about something
– The story concerns the prince's efforts to rescue Pamina.
– The book is primarily concerned with Soviet-American relations during the Cold War.
– This chapter concerns itself with the historical background.
– One major difference between these computers concerns the way in which they store information.3 worried and feeling concern about something
– Concerned parents held a meeting.
– concerned about/for something The President is deeply concerned about this issue.
– concerned for something He didn't seem in the least concerned for her safety.
– concerned (that)… She was concerned that she might miss the turning and get lost.4 concerned (about/with something) interested in something
– They were more concerned with how the other women had dressed than with what the speaker was saying.
Best Answer
The original (and still core) sense of concern has its roots in Latin con = with and cernĕre = to separate (which also give us discern = distinguish, recognise with the dis = apart prefix).
That's to say, concern = pertain to, be about, relate to - effectively, OP's definition #2.
In fact, OP's initial definition superficial = not [concerned with anything] serious or important would seem perfectly adequate to me even with the [bracketed] portion removed entirely (it adds little extra meaning).
Outside of additional context, "This problem concerns me" could have several meanings, including...
I hope that series of definitions gives some idea of how the meaning has become "stretched" over time.
The sentence "X is superficial" (from French superficiel = located at or on the surface) normally means...
Thus although I said above that definition #2 applies for OP's purpose here, it's also possible to say that definitions #1 and #4 could apply...
The one that hardly applies at all is #3. We wouldn't normally use "John is superficial" to mean John isn't anxious about, or upset by, important things (we'd say unflappable, nonchalant, even-tempered, etc.).