This is an interactive tool, a modal operator that draws the x, y, width and height of the bounding box corners viewed from front (use ortho to check):

    import bpy
    import bgl
    import blf
    from mathutils import Vector
    from bpy_extras.view3d_utils import location_3d_to_region_2d
    
    class G:
        top_left = Vector()
        bottom_right = Vector()
    
    def draw_poll(ob):
        if ob is not None and ob.type in ('MESH', 'CURVE', 'SURFACE', 'META', 'FONT'):
            return True
        return False
    
    def draw_callback_view(self, context):
        
        ob = context.object
        if not draw_poll(ob):
            return
        
        mat = ob.matrix_world
        
        # 50% alpha, 2 pixel width line
        bgl.glEnable(bgl.GL_BLEND)
        bgl.glColor4f(1.0, 0.0, 0.0, 0.5)
        bgl.glLineWidth(2)
        
        bbox = [mat * Vector(b) for b in ob.bound_box]
    
        bgl.glBegin(bgl.GL_LINE_STRIP)
        bgl.glVertex3f(*bbox[0])
        bgl.glVertex3f(*bbox[1])
        bgl.glVertex3f(*bbox[2])
        bgl.glVertex3f(*bbox[3])
        bgl.glVertex3f(*bbox[0])
        bgl.glVertex3f(*bbox[4])
        bgl.glVertex3f(*bbox[5])
        bgl.glVertex3f(*bbox[6])
        bgl.glVertex3f(*bbox[7])
        bgl.glVertex3f(*bbox[4])
        bgl.glEnd()
        
        bgl.glBegin(bgl.GL_LINES)
        bgl.glVertex3f(*bbox[1])
        bgl.glVertex3f(*bbox[5])
        bgl.glVertex3f(*bbox[2])
        bgl.glVertex3f(*bbox[6])
        bgl.glVertex3f(*bbox[3])
        bgl.glVertex3f(*bbox[7])
        bgl.glEnd()
        
        min_x = min(b.x for b in bbox)
        max_x = max(b.x for b in bbox)
        min_y = min(b.y for b in bbox)
        max_y = max(b.y for b in bbox)
        min_z = min(b.z for b in bbox)
        max_z = max(b.z for b in bbox)
        
        bgl.glColor4f(0.0, 1.0, 0.0, 0.5)
        bgl.glBegin(bgl.GL_LINE_STRIP)
        bgl.glVertex3f(min_x, min_y, min_z)
        bgl.glVertex3f(min_x, max_y, min_z)
        bgl.glVertex3f(max_x, max_y, min_z)
        bgl.glVertex3f(max_x, min_y, min_z)
        bgl.glVertex3f(min_x, min_y, min_z)
        bgl.glVertex3f(min_x, min_y, max_z)
        bgl.glVertex3f(min_x, max_y, max_z)
        bgl.glVertex3f(max_x, max_y, max_z)
        bgl.glVertex3f(max_x, min_y, max_z)
        bgl.glVertex3f(min_x, min_y, max_z)
        bgl.glEnd()
        
        bgl.glBegin(bgl.GL_LINES)
        bgl.glVertex3f(max_x, min_y, min_z)
        bgl.glVertex3f(max_x, min_y, max_z)
        bgl.glVertex3f(min_x, max_y, min_z)
        bgl.glVertex3f(min_x, max_y, max_z)
        bgl.glVertex3f(max_x, max_y, min_z)
        bgl.glVertex3f(max_x, max_y, max_z)
        bgl.glEnd()
        
        G.top_left = Vector((min_x, min_y, max_z))
        G.bottom_right = Vector((max_x, min_y, min_z))
        
        bgl.glColor4f(1.0, 0.9, 0.0, 1.0)
        bgl.glPointSize(5)
        bgl.glBegin(bgl.GL_POINTS)
        bgl.glVertex3f(*G.top_left)
        bgl.glVertex3f(*G.bottom_right)
        bgl.glEnd()
        
        # restore opengl defaults
        bgl.glLineWidth(1)
        bgl.glDisable(bgl.GL_BLEND)
        bgl.glColor4f(0.0, 0.0, 0.0, 1.0)
    
    
    def draw_callback_px(self, context):
        
        ob = context.object
        if not draw_poll(ob):
            return
        
        font_id = 0
        blf.size(font_id, 20, 72)
        
        region = context.region
        rv3d = context.space_data.region_3d
        
        text = "%.2f / %.2f" % G.top_left.xz[:]
        dim = blf.dimensions(font_id, text)
        loc = location_3d_to_region_2d(region, rv3d, G.top_left)
        loc.x -= dim[0]
        blf.position(font_id, *loc.to_3d())
        blf.draw(font_id, text)
        
        text = "%.2f / %.2f" % tuple(map(abs, (G.top_left - G.bottom_right).xz))
        dim = blf.dimensions(font_id, text)
        loc = location_3d_to_region_2d(region, rv3d, G.bottom_right)
        loc.y -= dim[1]
        blf.position(font_id, *loc.to_3d())
        blf.draw(font_id, text)
    
    
    class ModalDrawOperator(bpy.types.Operator):
        """Draw a line with the mouse"""
        bl_idname = "view3d.modal_operator"
        bl_label = "Simple Modal View3D Operator"
    
        def modal(self, context, event):
            context.area.tag_redraw()
    
            if event.type in {'ESC'}:
                bpy.types.SpaceView3D.draw_handler_remove(self._handle_view, 'WINDOW')
                bpy.types.SpaceView3D.draw_handler_remove(self._handle_px, 'WINDOW')
                return {'CANCELLED'}
    
            return {'PASS_THROUGH'}
    
        def invoke(self, context, event):
            if context.area.type == 'VIEW_3D':
                # the arguments we pass the the callback
                args = (self, context)
                # Add the region OpenGL drawing callback
                # draw in view space with 'POST_VIEW' and 'PRE_VIEW'
                self._handle_view = bpy.types.SpaceView3D.draw_handler_add(draw_callback_view, args, 'WINDOW', 'POST_VIEW')
                self._handle_px = bpy.types.SpaceView3D.draw_handler_add(draw_callback_px, args, 'WINDOW', 'POST_PIXEL')
    
                context.window_manager.modal_handler_add(self)
                return {'RUNNING_MODAL'}
            else:
                self.report({'WARNING'}, "View3D not found, cannot run operator")
                return {'CANCELLED'}
    
    
    def register():
        bpy.utils.register_class(ModalDrawOperator)
    
    
    def unregister():
        bpy.utils.unregister_class(ModalDrawOperator)
    
    if __name__ == "__main__":
        register()
        
        for area in bpy.context.screen.areas:
            if area.type == 'VIEW_3D':
                context = bpy.context.copy()
                context['area'] = area
                bpy.ops.view3d.modal_operator(context, 'INVOKE_DEFAULT')
                break

The relevant part for you (front view only):

    import bpy
    from mathutils import Vector
    
    ob = bpy.context.object
    mat = ob.matrix_world
    bbox = [mat * Vector(b) for b in ob.bound_box]
    
    min_x = min(b.x for b in bbox)
    max_x = max(b.x for b in bbox)
    min_z = min(b.z for b in bbox)
    max_z = max(b.z for b in bbox)
    
    top_left = Vector((min_x, max_z))
    bottom_right = Vector((max_x, min_z))
    
    # x and y
    #top_left
       
    # width and height
    #tuple(map(abs, (top_left - bottom_right))

And here's some code to write out what you asked for (front view):

    import bpy
    from mathutils import Vector
    
    def main():
        
        # specify the object
        ob = bpy.context.object
        
        # change to your desired output location and file name
        filepath = r"D:\temp\bounds_2d.txt"
    
        scene = bpy.context.scene
        frame_current = scene.frame_current
        frame_start = scene.frame_start
        frame_end = scene.frame_end
        
        write_bounds_3d(filepath, ob, frame_start, frame_end)
        
        scene.frame_set(frame_current)
    
    
    def write_bounds_3d(filepath, ob, frame_start, frame_end):
        
        file = open(filepath, "w")
        
        for frame in range(frame_start, frame_end + 1):
            bpy.context.scene.frame_set(frame)
            file.write("%f %f %f %f\n" % get_bounds(ob))
            
        file.close()
    
    def get_bounds(ob):
        mat = ob.matrix_world
        bbox = [mat * Vector(b) for b in ob.bound_box]
        
        min_x = min(b.x for b in bbox)
        max_x = max(b.x for b in bbox)
        min_y = min(b.y for b in bbox)
        max_y = max(b.y for b in bbox)
        
        top_left = Vector((min_x, max_y))
        bottom_right = Vector((max_x, min_y))
        
        # make flat tuple, x y width height
        return top_left.to_tuple() + tuple(map(abs, (top_left - bottom_right)))
    
    if __name__ == '__main__':
        main()