Rules As Written, It Shines Real Sunlight
Sunlight is a very specific word to use in both texts. They chose to use that word to interact with certain creatures like Banshees and Vampires. Paladins are famously effective against undead in many ways. This is one of those ways. By contrast, the Light cantrip only shines light, not sunlight. Sunlight is an established mechanic of Dnd5e
Another contrasting example, "Avenging Angel" emits an "aura of menace". In the absence of menace having meaningful mechanical impact, an aura of menace doesn't mean much, other than to describe what is causing the effects of Avenging Angel. But in the context of sunlight having a mechanical impact on creatures, an aura of sunlight is meaningful. The sunlight is causing the radiant damage. If menace had further implications, like if a creature was particularly sensitive to emotions, menace is a very important descriptor. The sunlight being magical is of no consequence, since both light and darkness can be magical and have the same effects as their natural counterparts.
Another example, look at Sun Blade
The sword's luminous blade emits bright light in a 15-foot radius and dim light for an additional 15 feet. The light is sunlight.
It should also be noted that in any case where the phrase "The light is sunlight" comes up, the preceding sentence describes mechanically what the light is doing in terms of bright and dim light. "Dim sunlight" and "bright sunlight" are not established phrases like dim light and bright light are, so sunlight is left out of the sentence. It's just for simplicity's sake that they reuse common sentences throughout the PHB to describe common effects like light. That's why the phrase "The light is sunlight" is appended rather than part of the sentence explaining light. In a similar way, the sunlight descriptor precedes the mechanical description of dim and bright light. Consistency is something the rules strive for, but occasionally it falls short, so the lack of the phrase "it is sunlight" is not enough of an argument to suggest that it isn't sunlight.
Finally, let us look at an example where the lack of specific description would indicate that it is NOT a specific type of light because the specific type of light is mechanically unimportant. Look at Moonbeam.
A silvery beam of pale light shines down in a 5-foot radius, 40-foot-high Cylinder centered on a point within range. Until the spell ends, dim light fills the Cylinder.
Moonbeam does not describe itself as being moonlight. It doesn't do this because moonlight is not an established mechanic in Dnd5e. Indeed, werewolves in 5e do not interact with moonlight in any way shape or form, contrary to our cultural understanding of what a mythological werewolf might be. What's special about Moonbeam is that it explicitly anticipates such an interaction with a creature like a werewolf.
A shapechanger makes its saving throw with disadvantage. If it fails, it also instantly reverts to its original form and can't assume a different form until it leaves the spell's light.
In this case, moonbeam has a specific interaction with shapechangers, but shapechangers do not have a specific interaction with moonlight. Sunlight does not have a specific interaction with anything, but creatures do have a specific interaction with sunlight.