The key to your problem is as follows: key() function selects matching nodes from a document the context node belongs to. In your case it's /categories/category/element element. If you want to select nodes from original document you need to switch context. Usually it's done using xsl:for-each:

<xsl:variable name="root" select="/"/>

<xsl:for-each select="document('categories.xml')/categories/category">
            <tr  bgcolor="silver">
                <td><xsl:value-of select="@name" />
                </td>
            </tr>
            <xsl:for-each select="element">
                <tr><td>

                   <xsl:variable name="element-name" select="@name"/>

                  <xsl:for-each select="$root">
                       <xsl:apply-templates select="key('animal-name', $element-name)" mode="test"/>

                 </xsl:for-each>
                     </td>
                 </tr>
            </xsl:for-each>
        </xsl:for-each>

Sure, but then you need to iterate over source elements, not categories - just select each element having no category: