Adding unique ID for groups of items using field calculator in QGIS

You can use arrays for this purpose:

 array_find(array_distinct(array_agg("Name")),"Name")

Which returns for example this id column:

So how does it work: First an array of all Name values is created, then every duplicate value gets deleted. At the end the index of the current name is searched in the array and the index used as id.

PS: not sure when array_find and array_distinct were introduced in QGIS. Only tested in 3.14. If you dont want zeros as id, simply add +1 to the expression.

You can use pyqgis:

lyr = QgsProject.instance().mapLayersByName('lmv ok_my_riks_sample')[0] #Change name to match your data
category_field = 'kategori' #Same
field_to_calculate = 'cat' #Same. This integer field needs to be added before executing the code

unique_vals = lyr.uniqueValues(lyr.fields().indexFromName(category_field)) #Find all unique values
#{'Sankmark', 'Tätort', 'Skogsmark', 'Vattenyta', 'Öppen mark'}

d = {cat:e for e,cat in enumerate(unique_vals, 1)} #Create a dictionary of unique values and number #Create a dictionary of unique values and number
#{'Sankmark': 1, 'Tätort': 2, 'Skogsmark': 3, 'Vattenyta': 4, 'Öppen mark': 5}

#Update field using dictionary
with edit(lyr):
    for feat in lyr.getFeatures():
        feat.setAttribute(feat.fieldNameIndex(field_to_calculate), d[feat[category_field]])
        lyr.updateFeature(feat)

This is a typical Case..When..Then solution. In the field calculator, you can use the following statement to update the ID field:

Case
When “Name” = ‘loam’ then 1
When “Name” = ‘silt’ then 2
When “Name” = ‘sand’ then 3
.
.
Else 20
End

You can add other cases for other names and assign them ids as you like. the Else is to provide one value for the remaining names.