In the cut scenes for Mipha, Daruk, and Revali it is made clear that each of their respective powers (healing, protection, upwards draft) is an ability unique to them (if not genetic as Daruk's offspring also have his ability of protection). When clearing the Divine Beast's their spirits transfer these gifts to you. However, Urbosa's divine beast cut scenes – and the way the Gerudo speak of the divine beast – indicate that the divine beast can generate lightening – not Urbosa.
So, how can she give you the power of lightning when it was not something innate to her like Daruk's protection?
Best Answer
Since the before-hand cut scenes only go into so much explanation about who the champions are or were, we can only get a few small pieces of information as to who they were as people. It's been 100 years and most of their abilities are either legend or forgotten.
For Mipha, it seems that she just has a natural ability to heal. That could mean that she was born to have healing ability out of the gate without any bloodline telling her she could, since her brother exhibits none of this. Neither does her father. We can take from that the idea the Mipha's is special to her.
Daruk obviously has his from a blood line. His entire family has had this power and will always have this power. His ancestor has it, so we know it travels down generations. Daruk's parents and so forth must have had it too. From that, we can take that it's a different source than Mipha's.
Revali doesn't seem to have a magical ability through blood line or special gift. His seems more like he's honed and trained himself to have this special ability. He brags about it like he's been able to master a technique that no other Rito has been able to emulate. In this way, it seems his power is more a hidden talent or trained magic.
So, looking at what we know about Urbosa, a number of theories could be created. Maybe her Divine Beast gave her the ability to unleash lightning so she could better control it. Maybe she just naturally was able to contain the Divine Beast's power and can command it's lightning at will. We know from previous Zelda games that Farore's element is wind. Wind is close to lightning in most elemental cases, which also explains things like the Ether medallion in Link to the Past. And since Farore's dragon is also present with lightning, we could infer that Urbosa is blessed by one of the goddesses, which is why she's not scared of the lightning. She doesn't fear it, because Farore protects her (even if she doesn't know who Farore is, really).
Not enough to go on, but since I believe each champion has their own sources for their powers, Urbosa's could simply be another facet of power that we never see the source of, because it doesn't really matter for the context of the rest of the story.