To be seeking asylum suggests the recognition that we need a sacred space. In the old-fashioned sense an asylum was an institution, so it may be that we are seeking safety in structure and proscribed boundaries.
Psychological / emotional perspective: An enclosed space where we have to abide by rules imposed by others can give a sense of safety and security. Seeking asylum where one is not living in danger from our peers and compatriots suggests we are conscious that our thoughts and beliefs are not in tune with others. This may reflect a working situation.
Material aspects: When an asylum seeker appears in dreams it is probable that we are neglecting part of our persona and putting it at risk.