And just in case you hadn't got it ( gotten it?) the first two times: 'Got' is British English, 'Gotten' is American.
Or, at least that's the case now. A few hundred years ago, the two forms of the past participle were widely used on both sides of the Atlantic. In the 18th century, British people and American people used both 'got' and 'gotten' interchangeably.
A relic of 'gotten' in British English is the phrase 'ill-gotten gains', and a relic of 'got' in American English is the ubiquitous 'I've got..'. British people are quite happy saying 'I've got [something]' in the present tense as an informal alternative to 'I have [something], Americans less so. 'I've got' sounds less correct to the American ear than to the British one. But is it wrong? Well, no. It's just different. 'Got' in this phrase isn't the same as 'get' as a main verb, meaning obtain, fetch, receive, become and so on. 'I've got a plan' is an idiomatic usage which doesn't fit into the get-got-gotten pattern, but that doesn't make it incorrect. It's just an exception to the rule.
'Got' for British English. 'Gotten' for American English.
Both are correct
Got in British English
Gotten in American English
There's no 'should' about it, Ivan! That's not how it is. As I said, 'I've got' in the sense of 'I have' is an idiomatic usage which is different from the usual meaning of the verb 'to get'.
'I've got brown eyes' is correct. This might look like a present perfect, but in fact it is exactly the same as 'I have brown eyes'. 'I've got' in this phrase has a present tense meaning. It doesn't refer to the action of obtaining something.
This is different from 'I've gotten some milk' (US English), which refers to an action - it means that you've just been to the store and bought some milk. You can't say 'I've gotten brown eyes', because this would suggest that you went somewhere and acquired them, which is very unlikely!
Oh, absolutely! :)