Frontend Form Search Users based on Tags
In order to get at the tag relationships, I think you'll need to use the relatedTo parameter instead of search. You'll still use search for the full-name search. This should solve both your problems.

The tricky part is fetching the appropriate elements to pipe into relatedTo() - you can't just send through the raw title string from the post data - you'll need to fetch the actual element ids.

Here's a very rough example. You'll need to replace 'facetFieldHandle_n` with the field handle names of your tag selector fields held against the users.

{% set facet1 = craft.tags.title(searchSkills).ids() %}
{% set facet2 = craft.tags.title(searchSchool).ids() %}

{% set params = ['and',
    { targetElement: facet1, field: 'facetFieldHandle_1' },
    { targetElement: facet2, field: 'facetFieldHandle_2' }
] %}

{% set users = craft.users.relatedTo(params).search(searchName).orderBy('score').all() %}

&lt;ul&gt;
    {% for user in users %}
        &lt;li&gt;{{ user.fullName }}&lt;/li&gt;
    {% endfor %}
&lt;/ul&gt;