Getting the Line or MultiLine coordinates in QGIS through PyQGIS

For multipart linestring you can use the following code (I tested it in 3.16.1)

layer = QgsProject.instance().mapLayersByName("folyo")[0]
for feature in layer.getFeatures():
    for part in feature.geometry().asMultiPolyline():
        print("part")
        for pnt in part:
            print(pnt.x(), pnt.y())

For single part linestring:

layer = QgsProject.instance().mapLayersByName("Single parts")[0]
for feature in layer.getFeatures():
    for pnt in feature.geometry().asPolyline():
        print(pnt.x(), pnt.y())

asMultiPolyline and asPolyline returns list of Points, so you can use indexes.

UPDATE to check layer geometry type use geometryType method of layer. To separate single and multi types use isSingleType function. Here is the code for it:

layer = QgsProject.instance().mapLayersByName("your_layer")[0]
if layer.geometryType() ==  QgsWkbTypes.LineGeometry:
    for feature in layer.getFeatures():
        geom = feature.geometry()
        if QgsWkbTypes.isSingleType(geom.wkbType()):
            # single
            for pnt in geom.asPolyline():
                print(pnt.x(), pnt.y())
        else:
            # multipart
            for part in geom.asMultiPolyline():
                print("part")
                for pnt in part:
                    print(pnt.x(), pnt.y())

If you want to extract the xy of vertices of a MultiPolyline geometry you can use the following code:

layer = QgsProject.instance().mapLayersByName("layerName")[0]
iter = layer.getFeatures()
for feature in iter:
    geom=feature.geometry()
    verts = geom.asMultiPolyline() # use geom.asPolyline() for line geometry
    for vert in verts:
        for i in range(len(vert)):
            print(vert[i].x(),vert[i].y())

The output will be like this

-3487202.783144876 10010395.30478035
-3428476.505393841 10049595.900699263
-3428476.505393841 10049595.900699263
-3374593.995628879 9923727.261801671
-3428476.505393841 10049595.900699263
-3359626.0983730364 10059037.052994486