I grew up IN the east coast vs ON the east coast

Solution 1:

According to the Oxford American dictionary:

Coast - The part of the land near the sea; the edge of the land.

This suggests that one grows up on the coast, rather than in the coast, since the latter would indicate being raised inside a piece of land. Usage statistics also support this conclusion, as "on the east/west coast" is significantly more popular than "in the east/west coast," according to Google Ngrams.

Solution 2:

It really depends on common usage around the headword. All of these are common usage:

I live in the east.

I live on the east coast.

I live in the east coast city of XXX.

I live on the outskirts of the east coast city of XXX.

I live in the suburbs of the east coast city of XXX.

We tend to use on for lines or borders between areas (hence on the border).

Note: I have a house on Lake Como (it's not actually on the lake; it's on the line around it).