The wall box is from the mid-1950's and my multimeter shows that it is grounded. There are a solid 120 volts when testing between live and ground. I noticed a bare copper wire coming from the cloth wiring in the back tied to a screw connected to the box. I looked around for usable screw holes in the box to screw a ground screw into but they were all way too large. Does this look okay? I get good continuity here between the green insulated copper ground wire I added and the rest of the box. The green wire will go to the grounding screw on the outlet.
Purely technically this is an unreliable connection. Think if your screw is a little conical, the wire can slide further from the screw and become loose.It can take weeks or months to be noticeable. There should be parts which prevent the the wire sliding between the surfaces. In developed countries there should be used certified connection boxes if one extends old wirings.
There should be parts which prevent the bending, rotating and pulling of the wire near the connection. Unprotected wires should be inside a closed space which is inaccessible without using tools.
Electrically acceptable grounding must be built to have low enough resistance. That can be true for the existing old grounding, but inserting wire length can be invalid due the maximum allowed resistance. A multimeter is useless for checking this.
Finally the job without license can be illegal even in case you can do it technically perfectly. Technically perfect is not purely reliable contacts, low enough resistance and good insulation & protection. There can be a regulation which states that old wirings aren't allowed to be continued, the whole branch should be rebuilt starting from the line breaker box and by using only currently valid parts.