It might be a symbiotic relationship. The black mamba benefits from Harry's magical protection and food provided by him, while Harry gains the snake's unique abilities like its speed, stealth and venomous attacks for his adventures and battles in the wizarding world.
Their relationship could be one of mutual respect. Harry would respect the thunderbird's instincts and power, and the thunderbird would respect Harry's leadership and magical abilities. They would rely on each other in difficult situations. For instance, in a battle against Voldemort's followers, the thunderbird might listen to Harry's commands to use its abilities in the most effective way.
Well, it might serve as a protector for Harry. Since black mambas are very dangerous and fast, it could quickly attack any threats to Harry, like Death Eaters. It could also use its stealth to gather information for Harry in the dark corners of Hogwarts or in the Forbidden Forest.
It could be a very protective relationship. Harry, having lost so many people he cares about, might be extremely protective of his cat. He would make sure it's always safe and well - cared - for, and in turn, the cat would be fiercely loyal to him, perhaps even putting itself in danger to protect Harry.
They can be combined by making the Black Mamba a part of the magical creatures in the 'Harry Potter' universe. For example, it could be a new kind of Animagus form.
In some fanfictions, he might find it in the Forbidden Forest during one of his adventures. Maybe while exploring the deeper parts of the forest that are off - limits to most students, he stumbles upon a young pegasus in need of help. Harry, being the kind - hearted person he is, bonds with the pegasus and it becomes his familiar.
Well, in this fanfiction, the tiger familiar may have special abilities. It could be able to sense danger before Harry does. So it would lead Harry away from places where there are traps or lurking enemies. Also, being a large and intimidating animal, it can act as a deterrent to some of the less brave adversaries that Harry might face, making it easier for him to get through certain areas or tasks.
One interesting 'Harry Potter' and 'Black Mamba' fanfiction could be one where Harry Potter discovers a magical Black Mamba in the Forbidden Forest. The snake might have special powers related to the dark arts that Harry has to either overcome or use to his advantage.
The phoenix might also be able to teach Harry in a way. Since it has its own kind of magic, it could show Harry things like how to control his magic better when it comes to difficult spells. It could demonstrate its own magic, like its ability to rise from ashes, and Harry could learn from that to be more resilient in his studies and face challenges in a new way.
Maybe through some kind of magical encounter in the Forbidden Forest. He could be on a special mission or adventure and stumble upon the Nemean Lion. The lion might sense that Harry has a pure heart and a great need for a powerful familiar, so it decides to bond with him.
They might have all sorts of wild and exciting adventures. Maybe they battle monsters or solve mysteries together.