What is the proper way to use time-series data for classification?
Welcome to the community!

Well ... the answer needs a bit of explanation. It is not about MLPclassfier or another python function. It is about background in Signal Processing/Analysis specially for bio-signal.

For different bio-signals like EEG there has been feature extraction studies for long while. For example in your problem, BCI, you probably know P300, etc which are common features (I am pretty outdated. You may have a look at recent review papers.). So usually you extract features from these signals and use them as the input of your classifier (those X, Y values that you mentioned). I explain a bit:

In General

There are general feature extraction methods from signals. They either extracted from signal in time-domain or frequency domain or time-frequency domain.

Have a look at those links and get an overview.

EEG

But for specific bio-signals you don't need to re-invent the wheel. There is a long history of methods for EEg signal analysis. For BCI in particular, there are again reviews to help you find good feature extraction and classification methods.

Finally my point is that, MLPclassifier is just a python function. According to my experience as a ML scientist coming originally from Biomedical Data Analysis, the question is more about "what to do?" before "how to do?". I hope it helped. If you had more questions please comment so I can update my answer.

Good Luck!