Clip raster data by multiple shapefile in ArcPy using Clip data management tool

The reason you're getting this error is because ListFeatureClasses does not accept a path as an argument, the wildcard parameter is used to filter the feature classes returned to only those matching a particular file name pattern, for example 'Edit*' will return only the feature classes that start with edit.

You need to set your arcpy.env.workspace first, this will be the location arcpy searches:

import arcpy, os

rasterlayer = ("L:ArcpyDataMosaic")  # file name needed here 
outputDir = ("L:ArcpyDataclip")    
arcpy.env.workspace = "L:ArcpyDatavector_dataselected_district"
arcpy.env.overwriteOutput = True
vector_layer = arcpy.ListFeatureClasses()

for vectordata in vector_layer:
    # the first part of the file name before extension (if present) then add .tif
    output = os.path.join(outputDir,os.path.splitext(vectordata)[0].split('_')[0] + '.tif')
    arcpy.Clip_management(rasterlayer, "", output, vectordata, "0", "ClippingGeometry", "NO_MAINTAIN_EXTENT")

Using the first part of the name of the input vector data spit by underscores and adding .tif to the end; ArcGIS doesn't require in many tools to stipulate the format, instead it takes the extension and loaction and works out the appropriate output driver.. if none is given the result is an Esri GRID which can cause issues due to strict requirements.