The word Messianic simply means,
“of Messiah”.  Those who worship and believe that eternal life and salvation are found only through Yeshua (Jesus), the Messiah, sent by G-d, are called Messianic.  Most Messianic Jews prefer not to call themselves Christians. 
The word Christian comes from Greek terminology and started in Antioch among Gentile believers.  Instead, they prefer more Hebraic terms, such as Messianic Jews.
Paul, as a Jew, simply preferred to say.
"I am a Jew."   Because Gentiles have been grafted into Israel, they are equal, adopted members of one family before G-d.  Some Gentiles call themselves Messianic Gentiles. 
In truth, we are all one people, one body,
and one bride in Yeshua our Messiah.  

Why Observe the Biblical Feasts?

Messianic believers observe the Biblical Jewish High Holy Days as described in Leviticus 23.  These celebrations and festivals were
determined by G-d, as spoken to the people through Moses, to be "celebrated as a
perpetual statute throughout the generations”.  Believers were never supposed to have stopped observing them.  They are a “moed”, a time to meet G-d.  G-d has appointed certain times to come and meet his people, and the feasts are those times.  There is blessing in them,
keeping them is a matter of obedience to
G-d’s written word, and in all ways,
each one points to Yeshua as Messiah.