Affine is a form of political, non-sexual, cross-species marriage (...which does not preclude traditional marriage for the participants).
- It is a cultural practice, an interstellar social institution.
Affinity is a relationship governed by obscure cultural rules concerning non-marriage bonding of individuals or families of different sophont species, and the associated inheritance of titles and property. The cultural conventions are comparable to individuals declaring themselves to be "blood-brothers" (or a gender-free blood-sibling concept), or to adoption of strangers into an extended family.
History & Background (Dossier)
Marriage can be quite a number of things to different populations. For some it has a religious connotations, for others a secular one. For some, the reproductive aspect of supporting species survival is paramount, while other emphasize the metaphysical aspects of emotional relationships and the hard-to-define concept of love. Many sophont species simply didn't have the human-model concepts of love and marriage in the Terran or Vilani senses. Human sociocultural models simply didn't work for those societies. 
As the early Third Imperium grew increasingly cosmopolitan with an increasing number of sophont species, the Imperial Nobility increasingly saw the need to tie together power blocs of different sophont species... and thus the concept of Affine became a resurrected reality. The practice of cultural affinity now mostly serves as a vehicle for tying together World-State Nobility, Gentlesophonts, Landed Nobility, and Imperial Clans so they can reach the highest levels of Imperial society. The Vilani and a number of earlier societies had their own cultural practices that were widely comparable to to the contemporary institution of cultural affinity. 
References & Contributors (Sources)
- Marc Miller. Agent of the Imperium (Far Future Enterprises, 2015), 155. (v1.1 - Annotated)
- Citation Missing - T5 Noble Patent 8.5" x 11" forms (reverse side) - corrected for T5.10
- Traveller Wiki Editorial Team
- Author & Contributor: Lord (Marquis) and Master of Sophontology Maksim-Smelchak of the Ministry of Science