While death and loss are a part of life, they're incredibly difficult to deal with. Being left behind (whether the leaving is voluntary or a choice) by someone important is something that many people and characters can worry about, even to the point of it becoming a fear that takes over their life.

This is one of the worst feelings to experience and it can be inflicted by anyone close to the character—a family member (parent, spouse, sibling, child), lover or romantic interest, best friend, mentor, etc. Someone who has experienced abandonment may develop a debilitating fear of it occurring again, but so can people who have never gone through it because they know the anguish it causes and don't want it to happen to them.

Whether it looks like guardedness or holding on too tightly, a fear of abandonment can manifest in a number of ways.

Maintaining shallow relationships (so the character never grows close to someone who could leave them)
Reluctance to fully commit to a relationship
Sabotaging promising relationships by pushing the other person away, treating them badly, cheating on them, abandoning them first, etc.
Believing that people are going to leave (due to insecurity, feeling unworthy of love, etc.)
Difficulty trusting others
Becoming possessive or manipulative as a way of controlling the other person and keeping them from close
Staying in an unhealthy relationship because the character believes it's better than being alone
Attaching too quickly to a partner, friend, etc.