Sounds like you have a finite element mesh with nodes and elements.
You can use ST_Relate with pattern 'FF2F11212' to pick out elements
that share the same edge. This DE-9-IM is sort-of a custom ST_Touches,
but only takes linear boundary overlaps. So if you have a table
"elements", and you want to find ones that touch ID 567:
FROM elements, elements as e
WHERE e.id = 567 AND
ST_Relate(elements.geom, e.geom, 'FF2F11212');