Didn’t find a documented tag for this.
There is 4 x amenity=shoe_shine in the database.
Together with kiosk perhaps? Tag:shop=kiosk - OpenStreetMap Wiki
shop=kiosk is for something completely different
Maybe you meant Tag:street_vendor=yes - OpenStreetMap Wiki ?
there are also shoe cleaning machines in some places like hotels. Has it something to do with it (a facility for cleaning shoes) or is it about a person offering goods or services on the street and shoe cleaning would be one possibility (type)?
The shoe shining business has disappeared in most “western” countries (replaced here and there by the machines mentioned by @dieterdreist), but it is still common in many other places of the world. I haven’t seen a permanent stand for this service like shown in the OP - so far I have just been in contact with the mobile shoeshining services in various places, but I agree to @chris66 that amenity=shoe_shine would fit. It could be combined with shelter=yes + shelter_type=sun_shelter.