Python 3, 62 47 57 bytes

Edit: - 5 bytes thanks to @Mr.Xcoder

Edit: +10 bytes to fix a bug

lambda x,y:({*"abc"}-{x[:1],y[:1]}).pop()+(x[1:]or y[1:])

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Python 3, 62 47 57 54 51 bytes

Edit: - 5 bytes thanks to @Mr.Xcoder

Edit: +10 bytes to fix a bug

Edit: -3 bytes thanks to @betaveros

Edit: -3 bytes by using max instead of pop

lambda x,y:max({*"abc"}-{x[:1],y[:1]})+max(x,y)[1:]

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Python 3, 62 47 bytes

Edit: - 5 bytes thanks to @Mr.Xcoder

lambda x,y:({*"abc"}-{x[:1],y[:1]}).pop()+x[1:]

Try it online!

Python 3, 62 47 57 bytes

Edit: - 5 bytes thanks to @Mr.Xcoder

Edit: +10 bytes to fix a bug

lambda x,y:({*"abc"}-{x[:1],y[:1]}).pop()+(x[1:]or y[1:])

Try it online!

Python 3, 62 57 bytes

Edit: - 5 bytes thanks to @Mr.Xcoder

lambda x,y:({*"abc"}-{x[:1],y[:1]}).pop()+x[1:]

Try it online!

Python 3, 62 47 bytes

Edit: - 5 bytes thanks to @Mr.Xcoder

lambda x,y:({*"abc"}-{x[:1],y[:1]}).pop()+x[1:]

Try it online!