For a specific tags list you can simply calculate the amount of tags from this list attached to each item using
SELECT item_id, SUM(tag_id IN (specific_tags_list))
FROM item_has_tag
GROUP BY item_id
To obtain the list of items which have all of the tags in tags list you must compare calculated SUM() with the amount of tags in the list:
SELECT item_id
FROM item_has_tag
GROUP BY item_id
HAVING amount_of_tags_in_list = SUM(tag_id IN (specific_tags_list))
items that item_has_tag.tag_id = 1 AND item_has_tag.tag_id = 2 AND item_has_tag.tag_id = 3 for example
For this case the query will be
SELECT item_id
FROM item_has_tag
GROUP BY item_id
HAVING 3 = SUM(tag_id IN (1,2,3))
PS. The fields pair (item_id, tag_id) must be defined as unique by index (in any order, maybe primary) in the table item_has_tag structure.
UPDATE
Your code works very well but I get into trouble when I try to get the result from the items list SELECT items.* FROM items, item_has_tag GROUP BY item_has_tag.item_id HAVING 3 = SUM(item_has_tag.tag_id IN (1,2,3)) Error "SELECT list is not in GROUP BY clause"
Variant 1:
SELECT items.*
FROM items, ( SELECT item_id
FROM item_has_tag
GROUP BY item_id
HAVING 3 = SUM(tag_id IN (1,2,3))
) subquery
WHERE items.id = subquery.item_id
Variant 2:
SELECT items.name /* , another fields */
FROM items, item_has_tag
WHERE items.id = item_has_tag.item_id
GROUP BY items.name /* , another fields */
HAVING 3 = SUM(item_has_tag.item_id IN (1,2,3))
Variant 3: disable session ONLY_FULL_GROUP_BY SQL Mode (not recommended).
PS. Variants 1 and 2 can be freely converted to JOIN form.