She thought the fact that Supergirl can fly could be a problem to regulatory agencies in the real world.
Actually some of these issues have been addressed in the comics. Some haven't but I think I can give answers to this, since I was a very experienced comic reader and a self-proclaimed know-it-all.
In the real world, she would probably need a locater beacon. It would more easily register her on radar and make her less of a flight navigation hazard. These devices are easily portable and could hang on her belt. Of course a supervillian could more easily find her this way too, but it's a small price to pay to insure lives in the air and she is the altruistic type.
Certifications for flying in general would be moot since she doesn't have to handle electronics or equipment to get airborne.
However she would have to file flight plans. A simple cell phone could be used so that she could enter a flight plan at really fast speeds. There's even a web site to do it fast. She just has to make sure the cell phone has internet access or even a text messaging system could be set up to accommodate her. If it's an emergency flight and she has no time to file a flight plan, she might decide to get deputized by the appropriate law enforcement agencies ahead of time. Cops are allowed to speed to rescues and ignore traffic laws, so why not her?
She would only need a specialized certification on how to avoid hazards and how to best file a flight plan.
As for drug testing, since she's invulnerable to needles and earth drugs couldn't effect her, the drug testing issue would probably be moot. However she would be required to voluntarily not fly if she feels her senses are impaired and she can be fined if she doesn't do it. Despite statutes in the US, if a law enforcement official feels a driver is impaired or unsafe to drive, a law officer can detain him even if his blood alcohol level is acceptable and there is no evidence of drug use. However, in other countries she might have more freedom to fly while impaired.
What could impair Supergirl? If she was affected by kryptonite or red sun radiation or super-extreme gravity like Krypton's gravity, if she got pummeled by a supervillian, or even a magic spell could affect her in this way. If she got into an accident while so impaired and she could have reasonably stopped herself from flying, she could be held accountable if she's caught flying while impaired.
Ignorance of known impairments is usually not an acceptable excuse
She would only be a UFO if she couldn't be identified (UFO=Unidentified Flying Object), so technically she wouldn't be a UFO in general. If Supergirl was willing to work with the appropriate agencies, they'd probably work with her. She might have to abide by no-fly zones in some countries though. However she did get UN sanction to deal with this.
However ljuvefreya forgot to mention that Supergirl would be a security risk. It is almost impossible to keep her out of anywhere she wants to go. Even though she is benevolent now, how could we defend against her if she ever went bad or got mind controlled by a supervillian? It's happened before to her and other superheroes. If this is a version of Supergirl that is affected by Kryptonite, then it shouldn't be a huge problem. In the comics government agencies have come up with effective ways to keep superbeings out of classified installations. Everyone has a weak spot even in fiction it seems.
There was a story where a cyborg Superman was trying to see President Clinton and the White House electronic defense system kept him out and almost killed him, but they identified him before any harm came to him.
The real problem is how do you make a superbeing behave so that they don't harm others? When in doubt, the same laws should apply to normal citizens as to superbeings in general. Supergirl is a US citizen, but she couldn't run for president because she wasn't born in America and she's not yet 35 years old, although she first appeared in 1959 in the comics, so in a way you might say she's 45 years old.
If she ever hurt someone accidentally or purposely, a defense could be that her special abilities were not reasonable to control amongst normal humans in such a situation. Then she would not be allowed to operate normally until some sort of control could be established so that it couldn't reasonably reoccur.
However she could be open to numerous nuisance suits dealing with sonic booms and accidental damage from low altitude flight. In the comics, superheroes almost always manage to fight in areas where the buildings are abandoned, but in the real world they would be liable for damage they could have reasonably avoided. If accidental damage was frequent, she could be incarcerated or special restrictions could be imposed to keep her from using any such powers she has that cause such damage and that may cause future damage. If she then defied such orders, she could be detained and even jailed.