This dream has, for some reason, stayed on my mind ever since I had it several weeks ago.

In this dream, I flew to the moon with some astronauts on the new Artemis rocket. I was not an astronaut myself, I was more like a guest or won the trip in a contest, or something of that nature. There were two of us "guests" on this trip. Myself, and a lady who very closely resembles someone who is a sister to an acquaintance of ours. I'm not saying it was her, but the resemblance is very strong. I don't know her and have only seen her very briefly. I've never really even thought much about her, for I am a happily married man.

So this lady and myself went through the training with the astronauts, and as we did, we began to fall for each other. We never told each other this, however. We walked on the moon together with one of the astronauts. I remember feeling how great it was to be there, and to experience something that almost no one else has. The dream then switched to after we had made it back to earth, maybe a couple of weeks afterward. I was sitting in a diner like restaurant with some family. The lady happened to be there working. I was getting up from the table to go tell her how I felt about her, and then I woke up.

So, last week, we went in a local grocery store that we don't go into that often, only to get certain items. The real life lady who is a very close resemblance was working in there - I believe that she is the store manager. She stared at me from the second we walked in, and that doesn't happen to me very often LOL. We ended up going through her checkout, She was very nice and friendly, and it was like she couldn't take her eyes off of mine. Again, up to that point, I've never really had any interaction with her at all.

Could it be possible that she had the same dream, from her point of view?

For this dream interpretation, I am using the ideas of the late Swiss psychologist Carl Jung.
The dreamer is in this dream.  He probably represents the dreamer’s ego.  For a Jungian, the ego is the center of consciousness.

If you are a male, this is a dream about your anima.  The anima is always female and is only found in the psyche of men.  The anima is capable of going through four stages of development. 
At the first stage, she is the primitive woman.  She sometimes appears in our dreams as a woman of another race.
At the second stage, she is the romantic lover.  She is usually an attractive young woman.
At the next stage, she is the good woman.  “The third is represented, for instance, by the Virgin Mary a figure who raises love eros to the heights of spiritual devotion.” P 185  I believe that a nun or religious woman, such as a Quaker woman, could also be symbols of the third stage.
At the fourth and final stage, “The fourth type is symbolized by Sapientia, wisdom transcending even the most holy and the most pure. Of this another symbol is the Shulamite in the Song of Solomon. … but in
this stage the eros has become spiritualized…. two examples of the fourth stage: the Greek goddess of wisdom Athena (left), and the Mona Lisa.”  P 185-186The identification of the stage of the anima is not always obvious (to me); however, she may very well be at the fourth and final stage of development.

The moon is a well-known symbol of the female while Artemis is the Greek virgin goddess of the hunt.  The lady, who went with the moon with the dreamer, is a symbol of the anima.The moon, Artemis and the female traveling companion area triad of the feminine.  The triple goddess is common in many religions/myths. 

The dreamer and his family were at a diner where this lady worked.  The dreamer’s family is probably symbolic of other personifications of archetypes from the collective unconscious.  In our dreams, eating food,being in a restaurant etc. is almost always a good sign.  In real life, food give us strength and helps us grow.  In our dreams, food helps us grow spiritually and psychology.  The lady works at this restaurant; therefore, she (the anima) is showing the dreamer’s growth and development.

Be sure to get back with me if this is too confusing.
Lugus