As far as investigated. It's not possible do that with google analytics until now! But here are some possibilities.
Web Tracking feature looks like help to track customer support. but it's in beta not for all "The User-ID feature is currently in closed beta and not available to all users.".
Still it's not possible to track the user personal id
User-ID enables the analysis of groups of sessions, across devices,
using a unique and persistent ID. For example, when used in
conjunction with an authentication system, a non-personally
identifiable ID representing a signed-in account could be sent to
Google Analytics as a User-ID.
GOOGLE ANALYTICS TERMS OF SERVICE
You will not (and will not allow any third party to) use the Service
to track, collect or upload any data that personally identifies an
individual (such as a name, email address or billing information), or
other data which can be reasonably linked to such information by
will comply with all applicable laws and regulations relating to the
collection of information from Visitors. You must post a Privacy
cookies that are used to collect traffic data, and You must not
circumvent any privacy features (e.g., an opt-out) that are part of
From above we can't make customer tracking using google analytics. But it will be very useful to have IMO.