There is no concept of "wallet ID" in Bitcoin transactions as transmitted over the wire. There are only input and output values in transactions (which do have an ID number).
If you know someone is connected to a given input, then you can trace them through any outputs to other parties.
So it's not quite as simple as a "wallet ID". The transactions in given Bitcoin wallet software may be connected to each other (by the mechanism I describe above), but are not necessarily connected, particularly if you generate a new address for each transaction for which you are the receiving party (and/or take additional precautionary measures). It's entirely possible to have a wallet full of untraceable transactions.
If you know someone is connected to a given input, then you can trace them through any outputs to other parties.
So it's not quite as simple as a "wallet ID". The transactions in given Bitcoin wallet software may be connected to each other (by the mechanism I describe above), but are not necessarily connected, particularly if you generate a new address for each transaction for which you are the receiving party (and/or take additional precautionary measures). It's entirely possible to have a wallet full of untraceable transactions.