I know what you're asking, how can this possibly be? How could a film which is full of animals have the same story as a romantic movie about a girl with hair the length of ten meters? Here's how.
First we take a look at the leading characters of the films. The females, Rapunzel and Judy Hopps, who are the main protagonists, are both underestimated. In Zootopia we see that nobody believes that Judy Hopps will be a good cop and similarly in Tangled, Mother Gothel tells Rapunzel she could never survive without her. The boys, Eugene and Nick, are outlaws. Nick is con artist and hated for being a fox and Eugene is thief. Rapunzel and Judy are both energetic and cheerful.
Look at the two images above. Look closely at the smiles and stance of each character. Nick and Eugene's smiles look familiar don't you think? They are both hunched over, making eye contact with their female counterparts. Rapunzel and Judy look in power in both images, Rapunzel's hand on her hip and Judy's hand in a fist standing strong.
So what are the plot similarities that lead to this idea. In Tangled, Rapunzel strikes a deal with Eugene to see the lanterns in exchange for the tiara, where as in Zootopia, Nick helps Judy in exchange for the recorder with his confession of tax evasion.
Eugene takes Rapunzel to the Pub where she feels uncomfortable; Nick takes Judy to a Naturalist club where everyone is nude in order to find out where Mr. Otterton went (Judy feels very uncomfortable).
Judy makes friends with Nick's enemy Mr. Big and clears his name; Rapunzel makes friends with Pub Thugs and later Maximus who was chasing Eugene. Also, Nick and Judy nearly escape through a toilet and fall into a river; Eugene and Rapunzel escape from a cave that is flooding and fall through a mini waterfall.
Throughout both films, relationships are developed, aswell as amazing similarities. It all begins with opening up - In Tangled, Eugene reveals his past to Rapuzel, Nick does the same in Zootopia. Both characters decide not to let others get to them. Rapunzel and Eugene achieve the dream of seeing the lanterns and try to become closer, eventually marrying one another; Judy and Nick find the fourteen missing predators and try to deepen their relationship (as cops, coworkers and as friends). Rapunzel thinks Eugene betrays her; Nick feels betrayed by Judy, however both female characters return home after feeling betrayed but when they find out the truth they both go back to the male protagonists.
Other similarities between the two films include; Mother Gothel telling Rapunzel, Eugene doesn't truly like her and that things will end badly; Chief Bogo tells Judy that she isn't cut out for the job and should quit.
Both films have a female villain who claim to care for the main characters but ultimately turn on them when the female characters resist: Mother Gothel "cared" for Rapunzel like a mother but once Rapunzel rebelled Mother Gothel tried enslaving her; Bellwether liked Judy since the two were both under appreciated, underestimated, and prey but once Judy tried to expose Bellwether's Night Howler scam she tried to kill Judy.
As you can see these films are ultimately alike. I suggest re-watching the films, having now read this article, to see if you can spot these similarities, trust me it's quite easy!