They are indeed illegal to my knowledge
Like I said, if you want to own a Thunderbird you'd have to create an explanation for how it got from North America to Europe, and it would have to be a well-kept secret - and the presence of a Thunderbird would be quite noticeable due to its size and the storms it summons. If any professors found out, it would surely be sent back to North America to be rehabilitated into its natural environment.
I recommend not owning one, because you'd have to explain how you keep it hidden - even in the Forbidden Forest, it would be pretty noticeable - but if you really really want one of your characters to have one and you're willing to put in all of the effort involved then what the heck.