Supergirl (2015– )
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Even if she is with people she knows at say home or work, she never knows when and if someone that doesn't know about her secret may pop by. So, she likely wears the glasses to be safe.
Another explanation could be, that the glasses complete the disguise of Kara Danvers, so she is used to wearing them when she's not Supergirl.
Also the lens of her glass prevents her from seeing with x-ray vision and you can notice it that she takes her glass halfway off when she want to use her x-ray vision when she is Kara Danvers. Edit
Yes and No. Supergirl is in what is called the "Arrowverse". The "Arrowverse" is a group of shows that share continuity and who's characters interact with each other across the different shows. However within the "Arrowverse" the character of Supergirl is from a different universe and a different Earth then the characters in the rest of the shows. Within the "Arrowverse" there are multipule universes that are all slightly similar to each other in different ways. Most of the shows take place in what they call "Earth 1", Supergirl takes place in another "Earth", but can come visit "Earth 1". Edit
There are three long-running iterations of the Supergirl character, the first being the original Kara Zor-El, Superman's cousin from Krypton, who died in the Crisis on Infinite Earths event, after DC Comics decided that Superman should be the sole Kryptonian survivor.
After continuity was rebooted, she was replaced by a protoplasmic, shapeshifter Supergirl also known as Matrix, created in another universe by that world's Lex Luthor. Matrix saved a girl named Linda Danvers and the two merged, becoming a new Supergirl. Linda eventually developed angelic powers, and after failing to save the original Kara Zor-El from her death, quit the Supergirl mantle.
After another DC universe reboot, the Kara Zor-El version of the character was re-introduced, and remains to this day.
Kara also has an Earth-2, more matured version of herself, Kara Zor-L/Power Girl. Edit
This is answered in season one episode seven Alex confronts Henshaw about the death of her father, which she believes Henshaw may be responsible for. He reveals instead that he is not, in fact, the Hank Henshaw who had been her father's partner. Henshaw and Mr Danvers had been tracking an alien refugee with Henshaw intending to kill it. Danvers had tried to stop him and both ended up dead. Instead, the man who Alex knows as Hank Henshaw is the alien they had been tracking: J'onn J'onnz, the last surviving Martian.
In the comics the character is known as the Martian Manhunter and has been a long time member of the Justice League. Beginning in the late 1980s he was the group's leader at a time when they were sponsored by Maxwell Lord, who is an antagonist to Supergirl on the show. J'onn has powers which are similar to Superman's but also include shapeshifting (as seen in this episode) and telepathy.
His civilian identity as Hank Henshaw is a bit of misdirection, since Henshaw is a known Superman villain in the comic books. The ambiguous nature of the real Henshaw's death in the show leaves open the possibility that the character will return in something close to his comic book state.
In the comics, Henshaw first appears as the commander of a space mission which encounters a burst of unusual stellar radiation. In a pastiche of the origin of the Fantastic Four, the crew begin undergoing strange changes, which eventually lead to the decay of all of their bodies. Henshaw manages to transfer his mind into the Lexcorp computers and is then sent into space when his consciousness begins interfering with Earth's communication systems.
In space, Henshaw becomes obsessed with revenge on Superman, who he blames for the death of his crew, including his wife. He next appears during the "Death of Superman" arc when Superman was believed killed by the monster Doomsday. At the time there were several beings who claimed to be a resurrected version of Superman. Henshaw had managed to recreate a body based on Superman's genetic code, but with cybernetic replacements for some body parts. He eventually gained recognition as the "true" Superman, but it turned out that he was working in conjunction with the alien warlord Mongul.