Genome’s Romance (2019)

Similarities:
ML has characteristics of a house pet with an ominous potential for converting completely into that animal.
set in contemporary Korea
show the value of friendship

Differences:
GR ML associated with a dog; MtSB ML is associated with a cat
GR ML is human becoming animal; MtSB ML is cat becoming human
GR FL has major health issue; MtSB does not
GR = web drama; MtSB = regular drama

Both have main lead characters that have an allergic reaction to the touch of another human being. However, there is one person in the world that is specifically a match for them......