Why can I not use 'my lubimy' instead of 'Ja i on lubimy'?
Because it is not asking you for that translation, it is not asking for "we like milk" but for "I and he like milk" so you would not be able to use "my lubimy"
In a normal situation, would it be ok to replace 'Ja i on' with 'my'?
Yes as long as it is clear who you would be talking about then "my" can be used all the time.
Lovely! Dzięki bardzo za pomoc!