Tho it grew from soil line,it is producing 2 type of fruits this year:
The usual short neck ,main crop and long neck Breba like fruits,all on new wood.
So with fig tree anything is possible.
I should mention that long or short neck they are all delicious.
Here are pixies:

I think it's great when a man plants fig trees in his front yard. I am about to do the same.

Back on topic...can a tree produce a shoot and two figs at the same node, a main crop fig in current year and a breba next year?

Very nice looking plant

Yes ,Anything is possible,with fig trees.
If the new wood of this year,is alive next year and if the breba buds makes it trough Winter,(Which here rarely do),then it will make Breba on same piece of wood next year.!