Lets hope this can be something done regularly.
(I have wondered if going forward they should just build satellites that have thrusters with the ability to refuel, a refueling port, or to accept a fuel pod while in orbit? That and providing a surface for a refueling vehicle grapple or latch onto.)
Similar things are currently done. My son was part of a team that holds the current Guinness record for maneuvering satellites in close formation. He’s an astronautical engineer and specializes in close formation satellite operations.
He’s flown closer than the current record but if this mission is successful, I assume Guinness will re-evaluate the record.
The obvious place to grab a bird is it’s engine bell of the apogee kick motor if it has such a motor attached too it. The engine is already structurally stressed for delta V in its primary thrust direction. You can insert a probe up the center and inflate a ball like bladder inside to lock the probe to the engine from the inside then you have push and pull delta V plus attitude control as the engine is fixed rigidly to the main body by default. Pretty much every GEO bird has a apogee kick motor used to go from GTO to it’s GEO slot takes about 1800 M/S delta V so it’s not a small rocket engine at all.
LEO birds might have their own circularization motor or it could have been inserted by a second or third stage that then dropped away and deorbited like SpaceX does with its second stages or the Russians do with Fregat & Soyuz if that’s the case there should still be a docking ring where the stage was attached in the first place that’s where your grab points are going to be you design a compatible grabber ring with finger like grabbers on the rim to mate male to male or female to male depending on what part was left attached to your target bird. AI Stereoscopic 3D 8K cams , laser LIDAR, and gyros should do the link up all by itself. Space robots humping in orbit 😂