Usually exchanges are required to do this by AML/KYC laws. Ask them, but I suspect you will get an answer similar to: sorry, gotta keep that in case the government knocks.
IS IT SAFE???
Depends. Do you trust your government? Its nothing they can change. They can either follow the rules of the country they work in or risk to get shut down. I think an exchange that is following the rules is more trustworthy as they have less reasons to scam you.
You can avoid this in the future if you trade directly with someone else, e.g. by meeting in person. Prices are usually a bit higher (~ +5%) when meeting in person, but if privacy is a concern it might be worth it.
NO FUTURE I GOT BURNED BECAUSE THIS IS ALL TO COMPLICATED
Dont give up. This might be an expensive lesson but you might as well be able to recover your coins, or (re)gain control over them.
This is the wallet address you provided: 35mYvUjMqiAYeobVhLc85PtHCmZKGeZ9Bi
Everything hinges on this. WHERE did you get that address (the 35mYvUjMqiAYeobVhLc85PtHCmZKGeZ9Bi)?
A bitcoin address is sort of like a bank account number. You bought bitcoins and told them to send it to a particular bank account number. They sent the bitcoins, and now you are asking them where your bitcoins are. Well, they don't have them, because they sent them to the bank account number you gave to them. They are now in that bank account number. You need to remember where or how you created that bank account so you can log in and use the coins.
This address is indeed strange/interesting. It is not likely that its made with bitcoin core, which might bring us back to the bc.i account.
A little background: Addresses that start with a 3 are usually multi sig addresses. Most addresses have a private key that allows you to sign a transaction. The private key is related to an address and is all you need if you want to spend bitcoin. Usually we dont have to handle those ourself, we have wallets for that. Either locally or as a service online.
Multi sig addresses can be signed not only by one private key but by several private keys, like a joint account. There are different types, but the most common would be 2 of 3, which means there are 3 possible private keys and to issue a transaction 2 signatures are required. But 1 of 7 is equally possible as 3 of 7 or 4 of 4.
Thus its not something you just just create.
AFAIK blockchain.info does not support multi sig addresses currently. Was it maybe from some other service or from the shop you mentioned?