It's neither necessary nor redundant: не (слишком) много = not (too) many. It just softens the expression не много (not many). However, you can't just remove слишком from the original sentence due to particular word order: in your suggested version, the meaning has changed dramatically into: It's not me (it's somebody else) who has many things keeping memories of childhood. To keep the correct meaning without слишком, you need to change the word order. e.g.:
Не (слишком) много у меня памятных вещей из детства.
Памятных вещей из детства у меня не (слишком) много.
A similar Russian phrase which could replace не слишком много: не так уж много.
Не (так уж) много у меня памятных вещей из детства.
Памятных вещей из детства у меня не (так уж) много.
many/much
andtoo
, they're not interchangeable or synonymous, they each have their own distinctive meaning which is attested to in general purpose dictionaries... it must be the same in the rest of the languages you're familiar with as well, at least Indo-European... i still encourage you to do more independent learning