XML Search and Change in SQL Server XML datatype column

Use the modify method on your XML type column.

update [YourTable]
set [XmlColumn].modify('replace value of
    (/assessmentItem/responseDeclaration/@identifier)[1] with "RESPONSE"'
)
where /*your where criteria here*/

The above XQuery example assumes your XML schema has only one occurrence of assessmentItem and/or responseDeclaration

As mentioned in the other answer, you can use SQL Server replace value of expression to modify a value in XML data type. It has limitation though, that only a single value can be updated at a time (see the documentation : replace value of (XML DML) > Arguments > Expression1).

Another thing to note is that your XML data has default namespace. So you need to tell SQL Server's XQuery about that, for example :

declare default element namespace "http://www.imsglobal.org/xsd/apip/apipv1p0/qtiitem/imsqti_v2p2";

For demo, consider the following data in XML variable :

declare @xml XML = '<assessmentItem xmlns="http://www.imsglobal.org/xsd/apip/apipv1p0/qtiitem/imsqti_v2p2" 
        adaptive="false" identifier="VHXYZ" timeDependent="false" 
        title="ZonesSS" toolName="IBIS Export" toolVersion="1.0">
    <responseDeclaration baseType="identifier" cardinality="single" 
             identifier="VH221999_order_match_choice_list_10.RESPONSE">
        <correctResponse>
            <value>i2</value>
        </correctResponse>
    </responseDeclaration>
    <outcomeDeclaration baseType="float" cardinality="single" identifier="SCORE">
        <defaultValue>
            <value>0</value>
        </defaultValue>
    </outcomeDeclaration>
</assessmentItem>'

Demo Query :

--value before
;with xmlnamespaces(default 'http://www.imsglobal.org/xsd/apip/apipv1p0/qtiitem/imsqti_v2p2')
select @xml.value('(/assessmentItem/responseDeclaration/@identifier)[1]','varchar(max)')

--update operation
set @xml.modify('
declare default element namespace "http://www.imsglobal.org/xsd/apip/apipv1p0/qtiitem/imsqti_v2p2";
replace value of
    (/assessmentItem/responseDeclaration/@identifier)[1] with "RESPONSE"'
)

--value after
;with xmlnamespaces(default 'http://www.imsglobal.org/xsd/apip/apipv1p0/qtiitem/imsqti_v2p2')
select @xml.value('(/assessmentItem/responseDeclaration/@identifier)[1]','varchar(max)')

output : (the top one is the 'value before' and the other is the 'value after')

VH221999_order_match_choice_list_10.RESPONSE
RESPONSE