The details you provided indeed suggest Elisa shares similarities with the Amphibious Man, or impart, in the least, an uncertainty about her origins.
In a conversation with Joe Utichi from Deadline, Guillermo del Toro emphasized the aquatic nature of Sally Hawkins' character:
She (Sally Hawkins) had been writing her own story about a woman who didn’t know
she was a mermaid. She and del Toro found the coincidence
serendipitous. “It was so beautiful that we were on the same
wavelength,” del Toro says. “I asked her if I could use this idea that
she had scars on her neck that turned out to be gills. She allowed me
to use another detail she had, which was that the character used a lot
of salt to make the water in her bathtub habitable.” source
Moreover, in an interview with io9, del Toro implied that Elisa is indeed the counterpart of the Amphibious Man:
It (the Amphibious Man) is a river God. It’s not an animal. It’s a river God in the Amazon.
There was never another one. There was him and Sally Hawkins put on
Earth, and their entire existence they were going to each other. And
they didn’t know. She was found in a river. No body knows who her
parents were. She has these markings since she was a baby. source
The ambiguity of Elisa's nature was an artistic choice, however, meant to be open to interpretation. As del Toro states in the same article on io9, unlike the characters in his other films, he never fully developed the background story of both the Amphibious Man and Elisa:
There is no larger backstory.
“I write eight-page biographies for most of the characters in the
movies,” del Toro said. “I give them to the actors. But my story for
the creature is in the movie.”
“The creature largely remains an enigma,” he (co-author Daniel Kraus) said. “And I think
Guillermo and I were fairly simpatico with that. Pretty copacetic in
our ideas in that the creature be allowed to represent different
things to different people.” source