Precocious doesn't mean multi-talented. It means having developed certain abilities or inclinations at an earlier age than is usual or expected. It is often used disparagingly, perhaps because precocious children are often show-offs.
As Odiously means repulsively, "odiously precocious" is extremely critical. It's a surprisingly harsh way to describe children.
It seems there may be a slight divergence in US/UK usage.
Longman Dictionary of Contemporary English says this:
a precocious child shows intelligence or skill at a very young age, or behaves in an adult way –
sometimes used to show disapproval in British English
and Cambridge Dictionary says:
(especially of children) showing mental development or achievement much earlier than usual:
- A precocious child, she went to university at the age of 15.
A precocious child behaves as if they are much older than they are:
The ambivalence is shown clearly at Thesaurus.com where, of the 24 synonyms given, more than half (marked with an asterisk) are disapproving:
*cocky, intelligent, mature, advanced,
*aggressive, ahead of time, beforehand, ?bold,
*cheeky, developed, early,
In the UK, it seems to me, 'precocious' is so often followed by 'brat' that even 'precocious child' has become disapproving. But speaking, for example, of someone's precocious talent, or describing someone as a precocious pianist expresses no such disapproval.