The second /t/ is not lost. It’s there, but it merges with the following /s/ (which, itself, is voiced into a /z/ because of the intervocalic context). So in the end you have /pətitzanimasjɔ̃/.

The man actually pronounces the second t. Even though it is not as clear as the sounds from the beginning of the word, I distinctly hear it. If you play the video at half-speed, you can hear it being merged with the z sound of the s liaison, PETI-tZanimations.

In standard French, the t in that phrase is not silent.

With the e (or es as here) at the end, the t is pronounced, maybe not forcefully, but still there. as already stated, the liaison is more between the s and the following a, so the t sort of gets lost.

I did not watch the video.

It depends. In French, the masculine form of the word is petit (no e at the end). In this case, if the word after “petit” starts with vowel, the “t” would be pronounced (i.e. un petit arbre). If the word after “petit” begins with a consonant, the “t” would be silent (i.e. un petit gateau).

The feminie version (petite) does not have this peculiarity, since there is always a vowel (petite) after it.