Yes BIRDTAMs exist, as do SNOWTAMs etc. I do not know about ICAO but they are recognised by EUROCONTROL (not sure if they are just free text or have ASTERIX and/or ADEXP formats).
Do a search on the above terms.
Here is a link to a Eurocontrol EATCHIP pdf file relating to ab-initio training mentioning ( in section 5.5) SNOWTAM, ASHTAM, BIRDTAM and FLOWTAMs)as being in the required syllabus. (I do not know the formal status of this document.)
<a href="http://www.eurocontrol.int/eatmp/humfactors/docs/T5-HUM.ET1.ST05.2000-GUI-01.pdf" target="_blank">www.eurocontrol.int/eatmp/humfactors/docs/T5-HUM.ET1.ST05.2000-GUI-01.pdf</a>