Albion maybe a democracy in name, but it's one that is highly tilted in favor of an entrenched aristocracy who monopolize the positions of power. Children are not taught that all men and women are equal. High House members are born High House Members, and what makes them High House Members? It's their bloodline that makes them a "blue blood". As with many historical real world aristocratic societies, there is almost assuredly a belief that their very blood is superior, and thus the products of illicit sex with inferior bloodlines (such as "beasts" or wherever it is the warriorborn get their genetics) would be considered likewise inferior. That said, practicality can trump bloodline in certain instances. For example, if Bennedict were heir to House Lancaster then according to Gwen that "might" trump his mixed genetics when it comes to his eligibility as a suitable mate to a gal from another High House.
In the case of the warriorborn, there is a perception held by Bennedict's own cousin Gwen that they're too focused on the present, and lack the mental ability to consider the past and plan for the future the way members of the High Houses can and do. This perception is very likely shared by the other High Houses, and probably explains why it is considered "unseemly" for a High House to recognize one as a member, why they are not considered eligible as spouses for High House members, and why it is considered "outre" for a High House member to get romantically involved with one. Furthermore as per Major Aspira's thoughts, "if he had been born like Ciriaco (ie. a warriorborn) he wouldn't be a major in the Auroran marines" , and given that Ciricaco is only a sergeant, this further implies that the Aurorans likely don't consider the warriorborn to be good officer material, especially not high ranking ones. Whether or not there is any truth to these perceptions is up in the air, but a certain subtle prejudice certainly seems to exist as a result.