to share a table with someone: to be reclining at the same meal, and so to be counted as part of that person's company
At a formal meal in the Greek and Roman world, guests did not sit on chairs. They lay on their sides on low couches drawn up around the table, propped on one elbow with the free hand for the food. This word takes the ordinary verb for that posture, ἀνάκειμαι, and puts "with" on the front of it. So it means to be stretched out on those couches alongside someone else, which in plain terms is to be at the same meal as that person.
The small word "with" is what carries the weight. A meal in that world sorted people, and the company on the couches announced who counted as your equals. When Matthew and Mark report that tax collectors and outsiders were at table with Jesus, they are not describing furniture. They are saying that these people were being reckoned as his company, and that is precisely what scandalized the onlookers.
The two shades of the word are one scene viewed from two sides. As a plain verb it reports that certain people joined a host at his table. Used as a name for a group, it turns that same fact into an identity: "the ones at table with him," the guests whose good opinion Herod could not afford to lose, the diners who muttered about a woman being forgiven. That is why one Greek verb comes into English sometimes as an action and sometimes as a noun.
It belongs to a small family of verbs for the same posture, and the difference is who they have in view. Both ἀνάκειμαι and the older κατάκειμαι simply describe someone lying at table and say nothing about company, while συνεσθίω names the act of eating with a person rather than the shared couch.