In general, no there is not a standardized alert system (unless you count email...). Most subareas of time-domain astronomy that deal with alerts tend to have their own central clearing houses and alert distribution mechanisms. Most of the high energy/explosive (supernovae, novae, blackhole transients (stellar or massive)) transient community tends to announce things via Astronomers Telegram which have basically replaced the older IAU Circulars and the Central Bureau for Astronomical Telegrams for announcing things. More recently, those working on comet outbursts have also sent notifications via ATel e.g. the recent Comet 29P outburst.
The NEO community tends to coordinate through the Minor Planet Center NEO Confirmation Page (NEOCP) for newly discovered NEO candidates found by the surveys. In recent years, this has been supplemented by the JPL/CNEOS Scout system which performs risk analysis on the NEOCP candidates and sends out email alerts for close passing objects or where they may be issues with the measurements - follow-up and discussion on these is done via email.
The Gamma Ray Burst (GRB) community has had the GCN Circulars as previously mentioned which started around the end of Compton Gamma Ray Observatory and the start of BeppoSAX when better location of GRBs become possible. An issue with both ATels and GCN Circulars is that they are written in English text and are sent via email and are mainly designed for human consumption and reaction. This has been recognized as an issue for many years and lead to the development of the VOEvent 2.0 which became an IVOA Recommendation in 2011. Quoting from the Abstract of the document:
VOEvent  defines the content and meaning of a standard information packet for representing, transmitting, publishing and archiving information about a transient celestial event, with the implication that timely follow-up is of interest. The objective is to motivate the observation of targets-of-opportunity, to drive robotic telescopes, to trigger archive searches, and to alert the community. VOEvent is focused on the reporting of photon events, but events mediated by disparate phenomena such as neutrinos, gravitational waves, and solar or atmospheric particle bursts may also be reported. Structured data is used, rather than natural language, so that automated systems can effectively interpret VOEvent packets. Each packet may contain zero or more of the "who, what, where, when & how" of a detected event, but in addition, may contain a hypothesis (a "why") regarding the nature of the underlying physical cause of the event.
It's fair to say that adoption of VOEvent has been slow and limited, driven in part by a lack of easy-to-use libraries or toolkits, but usage is picking up with the LIGO/VIRGO gravitational wave alerts going out in VOEvent XML format (Alerts User Guide) and a standard for reporting Fast Radio Bursts. Sending of VOEvents from the LSST brokers is also planned and so the standard is likely to see greater adoption.