Just use

<xsl:variable name="date" select="$element/date"/>

The way you are doing it, you are creating a temporary document to hold a copy of the date element. The variable is a document node, and document nodes have no name.

I would need the $date to be the same node as $date2 (I want to use a "xsl:choose" in the $date variable).

Either use XPath if() instead:

<xsl:variable name="date" select="if (X) then $element/date else $something.else"/>

Or use xsl:sequence and an @as attribute

<xsl:variable name="date" as="element(date)">
   <xsl:choose>
     <xsl:when test="X">
      <xsl;sequence select="$element/date"/>
    </xsl:when>
    ....
</xsl:variable>

Michael Kay
Saxonica
mike@xxxxxxxxxxxx
+44 (0) 118 946 5893