Drag a painted Shape

One approach might be to convert each Shape to a GeneralPath as the Shape is added to the panel.

Now the dragging logic only needs to support a single class that implements the Shape interface:

import java.awt.*;
import java.awt.event.*;
import java.awt.geom.*;
import java.util.*;
import javax.swing.*;

public class ShapePanel extends JPanel
{
    private ArrayList<ColoredShape> coloredShapes = new ArrayList<ColoredShape>();

    public ShapePanel()
    {
        DragListener dl = new DragListener();
        addMouseListener(dl);
        addMouseMotionListener(dl);
    }

    @Override
    protected void paintComponent(Graphics g)
    {
        super.paintComponent(g);

        Graphics2D g2 = (Graphics2D) g;
        g2.setRenderingHint(RenderingHints.KEY_ANTIALIASING, RenderingHints.VALUE_ANTIALIAS_ON);

        for (ColoredShape cs : coloredShapes)
        {
            g2.setColor( cs.getForeground() );
            g2.fill( cs.getShape() );
        }
    }

    @Override
    public Dimension getPreferredSize()
    {
        return new Dimension(600, 500);
    }

    public void addShape(Shape shape, Color color)
    {
        // Convert the Shape to a GeneralPath so the Shape can be translated
        // to a new location when dragged

        ColoredShape cs = new ColoredShape(color, new GeneralPath(shape));
        coloredShapes.add( cs );
        repaint();
    }

    class ColoredShape
    {
        private Color foreground;
        private GeneralPath shape;

        public ColoredShape(Color foreground, GeneralPath shape)
        {
            this.foreground = foreground;
            this.shape = shape;
        }

        public Color getForeground()
        {
            return foreground;
        }

        public void setForeground(Color foreground)
        {
            this.foreground = foreground;
        }

        public GeneralPath getShape()
        {
            return shape;
        }
    }

    class DragListener extends MouseAdapter
    {
        private GeneralPath dragShape;
        private Point pressed;

        @Override
        public void mousePressed(MouseEvent e)
        {
            //  the clicked Shape will be moved to the end of the List
            //  so it is painted on top of all other shapes.

            if (SwingUtilities.isLeftMouseButton(e))
            {
                pressed = e.getPoint();

                for (int i = coloredShapes.size() - 1; i >= 0; i--)
                {
                    ColoredShape cs = coloredShapes.get(i);

                    if (cs.getShape().contains( pressed ))
                    {
                        coloredShapes.remove(i);
                        coloredShapes.add(cs);
                        repaint();
                        dragShape = cs.getShape();
                        break;
                    }
                }
            }
        }

        @Override
        public void mouseDragged(MouseEvent e)
        {
            if (dragShape != null)
            {
                int deltaX = e.getX() - pressed.x;
                int deltaY = e.getY() - pressed.y;

                dragShape.transform(AffineTransform.getTranslateInstance(deltaX, deltaY));

                pressed = e.getPoint();
                repaint();
            }
        }

        @Override
        public void mouseReleased(MouseEvent e)
        {
            dragShape = null;
        }
    }

    private static void createAndShowGUI()
    {
        ShapePanel shapePanel = new ShapePanel();

        GeneralPath barBell = new GeneralPath();
        barBell.append(new Ellipse2D.Double(100, 100, 75, 100), true);
        barBell.append(new Rectangle2D.Double(150, 125, 175, 50), true);
        barBell.append(new Ellipse2D.Double(300, 100, 75, 100), true);
        shapePanel.addShape(barBell, Color.RED);

        shapePanel.addShape(new Rectangle2D.Double(100, 300, 75, 150), Color.BLUE);

        Polygon triangle = new Polygon();
        triangle.addPoint(0, 0);
        triangle.addPoint(100, 0);
        triangle.addPoint(50, 100);
        triangle.translate(400, 250);
        shapePanel.addShape(triangle, Color.GREEN);

        JFrame frame = new JFrame("ShapePanel");
        frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
        frame.add(shapePanel);
        frame.pack();
        frame.setLocationByPlatform(true);
        frame.setVisible(true);
    }

    public static void main(String[] args)
    {
        java.awt.EventQueue.invokeLater( () -> createAndShowGUI() );
    }
}