Como dizer "personalidade própria" em inglês

This means teenagers get a personality all of their own.
Here we are talking about temperament, their charactersistic way of reacting in general, etc.

The household robot with a mind of its own.
Here an inanimate thing that we personify (describing it as a having animate characteristics) by way of imagery or metaphorical ways.

Individualistic people capable of accepting that a dog also possesses  it's own personality.

Each sunflower has it's own personality, faces turned out so cute. They all have a little of yarn hair. 

As you see the last two sentences has a ring of "individuality" (say what one stands out from the pack/from the crowd or from the other party.

I hope this helps.

Sometimes "to have a mind of his own" - to have strong opinion and have your own opinions without being influenced by other people.
That is, to be yourself (on your thinking) and to be a stubborn maverick (on the thinking of others).

' Own Personality '

> Science Says You Have a Pretty Good Sense of Your Own Personality;
> People are actually better at judging their own personalities;
> Your Opinions of Other People Reveal Your Own Personality Traits.