Easy to do with some python run in the QGIS console:
Copy and paste this code snippet in the editor of QGIS python console and change line as appropriate to fit your data
layer_to_update = QgsMapLayerRegistry.instance().mapLayersByName('point_layer_name')[0] #change as appropriate (name of your point layer)
layer_to_join = QgsMapLayerRegistry.instance().mapLayersByName('poly_layer_name')[0] #change as appropriate (name of your polygon layer)
field_to_fill = layer_to_update.fieldNameIndex('point_layer_field') #change as appropriate (name of the field in the point layer thaht you want to fill
layer_to_update.startEditing()
for point in layer_to_update.getFeatures():
gpoint = point.geometry()
list = []
for poly in layer_to_join.getFeatures():
gpoly = poly.geometry()
if gpoint.intersects(gpoly):
list.append(poly['poly_layer_field']) #change as appropriate (name of the field where are stored the name of your polygons
layer_to_update.changeAttributeValue(point.id(), field_to_fill, ', '.join(list))
This will fill the point layer field you choose with a comma separated list of the overlapping polygons name for each point as shown in the 'poly' field below: