Ok, then yes that's very doable. All you'll need to do is create the field to contain that value. So for example a Text fieldtype. Next you'll need CB Query Field plugin, which will provide the Query Validation functionality in order to validate the field against existing database values. So for example the below should work.
Title: Identification ID
Name: cb_identification
Type: Text
Required: Yes
Integrations > CB Query Field > Validation
Query Validation: Enable
Query:
Code:
SELECT `id` FROM `#__comprofiler` WHERE `id` != '[user_id]' AND `cb_identification` = '[value]'
Validate On: Empty Results
Ajax Validation: Enable
This should take the value they input during registration or profile edit and compare it to the field value of other existing users. If a result is returned from the query it will fail the fields validation. You may also want to use the below to restrict the field to alphanumeric characters only depending on what characters can existing in your ids.
Parameters > Validation
Validation Rule: Custom REGEXP Validation
Custom REGEXP Validation: /^[a-zA-Z0-9]$/