Yes, magical flight works like natural flight
The concept of "natural flight" is a relic from D&D 3.5e, and the Fly skill inherited it that wording. "Natural flight" simply means non-magical flight, but all rules that affect flight speeds will affect you regardless of how you obtained it, unless said otherwise.
This wording appears originally in the Flight universal monster ability, but the same applies to it. The current version in the bestiaries (as per Bestiary 6) makes no mention of "natural flight" or it's maneuverability, simply that supernatural flight doesn't work in areas of antimagic.
You will also notice that all other magical means of flight also grant a maneuverability type, like the Wings of Flying magic item, or the Wind Walk and Beast Shape spells. Or even say that when one isn't specified, to treat as average maneuverability. So, it is commonly accepted that magical flight is also affected by maneuverability types and gets its bonuses or penalties.
There is an open FAQ request asking to clarify this, but since it got so little votes, it got ignored for a decade. On the other hand, in a completely different context, they have said that the maneuverability bonus affects all flight forms (FAQ):
Flight and Magical Flight: Can a paralyzed or stunned creature keep flying with magical flight? Does a creature with magical flight not apply bonuses or penalties to Fly checks because it doesn’t have a “natural” fly speed? Does flying make a creature immune to being flat-footed?
No, any creature that loses all actions can’t take an action to attempt a Fly check to hover in place and thus automatically falls. That includes a paralyzed, stunned, or dazed creature. Magical flight doesn’t act any differently, even for paralysis, as it isn’t a purely mental action. A creature with 0 Dexterity can’t fly, and paralysis sets a creature’s Dexterity to 0. Despite the fact that the Fly skill mentions that bonuses and penalties from maneuverability apply to creatures with natural fly speeds, they apply for any fly speed. If they didn’t apply to creatures that gained flight artificially or through magic, then those maneuverabilities (like the listed good maneuverability for the fly spell) would have no game effect. Finally, the statement “You are not considered flat-footed while flying” means that flying (unlike balancing using Acrobatics or climbing) doesn’t automatically make you flat-footed or force you to lose your Dexterity bonus to AC; it doesn’t mean that flying makes you immune to being caught flat-footed.
Also, note that, while the skill first says that you must have a natural means of flying to be able to put ranks into it, in the next phrase it explains how you can put ranks into it if you are able to gain flight by magical means, such as being able to cast a spell every day. SO, that text isn't really excluding magical flight.
Finally, the rules mentioned in the Movement page of the (unofficial) SRD are not official, those rules are picked among the rulebooks and put together in that page in an attempt to organize and make sense of the movement rules in general. The text you quoted was never printed in any of the books.