Update name of the object based on text located in that object?

As requested, try this for polygons other than rectangles.  This should work for the text frame whose centre lies within the path (without depending on the bounding boxes).

var doc = app.activeDocument;
var items = doc.pathItems;
var text = doc.textFrames;

for (var i = items.length - 1; i > -1; i--) {
    for (var j = text.length - 1; j > -1; j--) {
        var textBounds = text[j].geometricBounds;
        var centre = [
            textBounds[0] + ((textBounds[2] - textBounds[0]) / 2), 
            textBounds[1] + ((textBounds[3] - textBounds[1]) / 2)
        ];
        if (pointIsInPoly(items[i].pathPoints, centre)) {
            items[i].name = text[j].contents;
            text[j].remove();
            continue;
        }
    }
}

// Jonas Raoni Soares Silva
// http://jsfromhell.com/math/is-point-in-poly
function pointIsInPoly(poly, pt){
    for(var c = false, i = -1, l = poly.length, j = l - 1; ++i < l; j = i)
        ((poly[i].anchor[1] <= pt[1] && pt[1] < poly[j].anchor[1]) || 
         (poly[j].anchor[1] <= pt[1] && pt[1] < poly[i].anchor[1])) 
        && 
        (pt[0] < 
         (poly[j].anchor[0] - poly[i].anchor[0]) * (pt[1] - poly[i].anchor[1]) / 
         (poly[j].anchor[1] - poly[i].anchor[1]) + poly[i].anchor[0]) 
        && 
        (c = !c);
    return c;
}