Since I was young, I have been collected manying things which I like.

'have been collecting' is more appropriate for collecting over a period of time, whereas 'have collected' makes it sound like the things were collected just once. As it is stated that the collecting has been happening since young, 'many' may not be necessary to make your point.

For example, I have cassettes that havecontaining many fairy tales.

I have many map books that I bought duringwhile traveling.

Since it's the 'I' who was 'traveling', 'while' would be more appropriate. 'during' could be used for an event that is phrased as a noun or is distant from the subject, for example: 'during the school festival'.

I even have a roly poly toy whichfrom when I was 6 years old.

The collections give me feeling a sense of ease and help me remember mythe good old days.