Omitting current entry from list of entries generated from a relationship
You probably kind of confuse your "apprenticeships categories" with your "apprenticeships" section. Try something like this instead:

{% set relatedEntries = craft.entries({
    section: apprenticeships,
    id: 'not ' ~ entry.id,
    relatedTo: {
        targetElement: entry.apprenticeshipsCatsField
    }
}) %}


You now want to order them by your categories? Have a look at my answer I just posted here: "How do I display entries, grouped by distinct categories...".