Is there a short term for lack of a date of expiry when talking about documents? I'm specifically interested in identification document (ID).
I would simply use the term non-expiring. For example, in Washington state, one can obtain a non-expiring license for child care.
You may also encounter the phrase valid in perpetuity, e.g. Hawaii Rev. Stat. §481B-13 (about electronic gift cards/certificates):
This phrase is generally used in describing the length of effect of a legal document (like a contract or a constitution) or financial instrument rather than an identification, though, so I would consider it a secondary choice.