What I wrote could not be less true.
I think that the answer "I like sports" is also a possibility, am I wrong? I am not sure since I am not a native English speaker
You're not wrong, but, "I like sport" a far more common way of saying it. It implies you like sport in general rather than a few specific sports.
I'm still getting 'I love sport' marked wrong April 2019.
Is it possible to tell this is Ukrainian or Russian by the accent?
I would say no.
Я люблю всі види спорту