Although it is a common trope these days, it was not so in the genre of mahou shoujo (magical girl) manga and anime before Sailor Moon. In the long-running genre, many of the classic magical girls looked very different in their transformed state as compared to their civilian state (for example, Maho no Mako-chan, Creamy Mami, Mahou no Yousei Persia, Magical Emi, Fashion Lala, etc.). Sailor Moon largely helped set a new standard that caught on in the genre (though there are still series which use the older classic style of very different looks for civilian vs. transformed forms, such as Pretty Cure does).
Judging by the mangaka Takeuchi Naoko's early concept art for the character designs, she had not intended them to look obviously identical in both forms. All of the sailor soldiers wore a mask of some sort so that, like Tuxedo Kamen, they would be hard to identify by enemies and bystanders. The early published artwork also featured a mask for [Sailor Moon] in the style of Sailor V's.
In the concept art, Sailor Moon was drawn with pink hair and Sailor Jupiter with olive-green hair; this may or may not have been the same shade as their civilian hair color: it would have matched the magical girl standard for Usagi's hair to change from her civilian white (platinum blonde) or blonde hair color into a different shade, such as pink (the most common and symbolic color for magical girls).
In terms of an in-universe explanation for the phenomenon, one is not given, but it is demonstrated that while people do not generally recognize them, there is not some magic that flatly prevents it, as there are some occasions in which someone does recognize a person who is transformed. In episode 183, Sailor Star Fighter looks at Sailor Moon and suddenly an image of Usagi flashes through her mind; she quickly shakes it off, saying that's impossible (she doesn't want to admit it because it is easier for her personally if Sailor Moon and Usagi are different people, so she willfully ignores what she noticed). As another example, in episode 190 Sailors Uranus, Neptune, and Pluto intentionally transform in front of Taiki Kou and Yaten Kou because they are sure that Taiki and Yaten are Sailor Star Maker and Sailor Star Healer, respectively. Neptune asks, "So you aren't easily surprised, are you?" and is answered with "We anticipated it, somehow."
In Sailor Moon Crystal, recognition seems to be easier than in the classic anime. Tuxedo Kamen seems to have no trouble identifying Usagi as the same person as Sailor Moon whenever she is transformed into her various guises using her Hensou Pen (Transformation Pen). When Rei sees Sailor Moon in episode 3, she quickly realizes it's Usagi. We could chalk these up to Rei's intuition or soldier abilities and Mamoru's powers, but even Usagi's father is able to notice her when she is in a transformed form in episode 4. She walks past him at a party in her transformed form and he does a double-take, thinking he saw his daughter, and only subsequently concludes that he must have made a mistake.