I have figured out the cause of this issue (at least for my case). I found that @tassones answer about using the "Finish Sketch" command worked, but it got me wondering why does ArcMap think that this sketch is unfinished? It is just a point after all and simply placing it on the map should finish it (it usually does work this way).
So I double-checked, and found that it was actually a MULTI-POINT feature class, not a normal point feature class. Ie, each feature can be multiple points.
So I then used the "Multipart to Single part" tool to convert the feature class and creating points in the converted feature class worked normally as expected.
Really, this should be considered as a bug in ArcMap. Instead of silently failing to save what may be an unfinished "sketch", ArcMap should either "finish" the sketch and then save, or ask the user what to do. Or at the very least, produce an error message stating that the sketch was aborted because it was not finished. Silently destroying data the user has just created is a terrible behaviour for any application.