Can RNN learn for each `t` in time from a whole new dataset (many entries) Basically, my data set is not as simple multi-variate time-serie as it's often (to some extent) the case.

For each month, I have N entries (not less than 3000). Can RNN of any variant (Please bear my ignorance, as I am a newbie in deep learning) catch in memory what it learnt from i.th subset and proceeding with learning ...

Data is like:

+======================================================================================================================================================================================+
| index, target, feature1, feature2, feature3, feature4, feature5, feature6, feature7, feature8, feature9, feature10, feature11, feature12, feature13, feature14, feature15, feature16 |
+======================================================================================================================================================================================+
| 2013-01-01, 1, 12, 0.006750, 21.192372, 39.119279, 0, 0, 0, 0, 13.602740, 117691.0, 0.06, 17259.0, 61491.0, 10.960000, 44620.0,                                                      |
+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| 2013-01-01, 1, 12, 0.256899, 21.192372, 39.119279, 0, 0, 0, 0, 30.282192, 835.0, 0.06, 221.0, 344.0, 10.004412, 406.0,                                                               |
+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| 2013-01-01, 0, 12, 0.000500, 21.192372, 39.119279, 0, 0, 0, 0, 30.282192, 49292.0, 0.04, 10853.0, 22945.0, 10.004412, 20132.0,                                                       |
+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+


and the list continues for other months.

EDIT

My baby wail data-frame, with repetitive multivariate times-series is like 

index        features...
2013-01-01  
2013-01-01
2013-01-01
...
2013-02-01
2013-02-01
2013-02-01
...


and the list goes on with the same length for each month.
As I want to learn from time along with other features, I reshaped the data-set with multi level indexing into subsets with the same length taking  the next nth*iteration entries and spreading over subsets tails.

I got something like 

multi_level_index   features...
1 2013-01-01
  2013-02-01
  2013-03-01
  ...   
2 2013-01-01
  2013-02-01
  2013-03-01
  ...
...


Can I benefit all learning data and feed into LSTM Keras solution (or other, but preferably in Python)?
I am following an approach I found here
So, from what I understand you are trying to predict the target column given all your feature columns and the information from the past. This feels like a time-series forecasting problem to me too (You can check variations online).

Vanilla RNN tends to vanish or explode the gradients especially given that you have such a large number of timestamps. I'd suggest you to try an LSTM in order to avoid it as much as you can. Ideally yes, it is supposed to remember what happened earlier and learn from that.