From what i have understand regarding crypto(I'm also quite a newbie in crypto), that will depend on the ICO's Token standard. For example, a certain ICO has ERC-20 token standard(which are common among new ICO's), which means your wallets should support an ERC-20 token for you to be able to transfer from both wallets. I'm not entirely sure as i'm still in the midst of learning about cryptos.
Im in the same boat guys! I have a wallet for Ether and ERC20 Tokens. Something I have learned is not all wallets that accept ERC20 will automatically show the token, in fact in many cases while the token is there, you cant see it until you have added the Contract Address. This allows the Wallet to highlight that the coins/tokens are there and shows you a balance.
To do this - go to Etherscan.io and search for your Wallet address that you have issued for people to send the coin/token to and look for transactions coming into your Wallet. It may default to show you Ether, but there is a drop down menu you can select to show Tokens. Regardless of what you see in your Wallet when you open it, if its shown on Etherscan.io as being sent to your Wallet then it is there - so check this before complaining to Devs that you didnt get the Airdrop.
Once you see the list of tokens sent it will give you a time stamp etc - click on the transaction which will open a Transaction Information page showing things like the Hash/Receipt(should say success if its been delivered to you), Block Height and sender. Among the other things is the Contract Address - copy that and paste it into the New Token section of your Wallet and that should be that.