As I'm sure most have noted...
Each person is different.
Therefore, they all feel certain emotions that only pertain to them.
We may feel similarly, but not exactly.
With different people comes different views.
Even an anime that has blood and gore can make the viewer laugh, indifferent, or left trembling.
Personally, how the director guides the anime, to the art, music, seiyuu, and other significant circumstances that influence it may lead to the overall feeling.
Most of the anime I have watched has left me fulfilled. I feel satisfied to finish them and I enjoyed the ride from start to finish.
Some anime leave me trolled or exasperated. It makes me wonder how the producers can showcase such a butchered-up anime.
If the scene is melancholic and has some emotional impact, perhaps I'll be crying or reminiscing.
As for current anime, most of them are decent. I don't feel much attachment to them probably because they haven't left much for me to remember or to go back to re-watch. Could be because I watched anime for a little over 17 years now and taking a slight break...
To each their own.
Depending on the anime, what we are feeling at the moment (whether from work, school, or from society), and how we view the anime (stream, download, on TV), may influence our mood.