In several Law systems, someone in custody and awaiting trial can chose who will be responsible for his material wealth. In some systems, the Government of the Victim party has the right to trial, in others the government of the Felon will want to have the trial. We had some diplomatic disputes over this. We don't what know what government will hold the trial. The Alliance isn't the Earth's Government, Earth still has nation states. Shepard could be from the U.S. or maybe he could be my Grandgrandgrand (you get the idea)child and be from Germany.
By the Way, i want Shepard eating a Bratwurst in ME3
EDI and the Normandy is something different, it will have to be determined if EDI is officially a sentient being and can belong to a Person.
Making an AI may be illegal, so would purchasing one be. Creating EDI will be added to the long list of Cerberus Crimes.
If EDI is found to be a sentient being, she may lay claim to the Normandy. Cerberus is a non recognized rogue element doesn't even have the right to exist, let alone own anything. Either they "go legal" and stand trial for everything or they don't and everything that falls into anyone's Hands wont be cerberus gear anymore since they are outlaws.
EDI could be seen as a sentient being and as part of the Normandy. It could be seen as a rather harsh form of imprisonment to get the bluebox out and deprive her, a sentient being, of all sensory input. This is considered the worst kind of torture by many people.
EDI hasn't committed any crimes that would warrant something that would be akin to scooping the part of your Brain out that processes sensory information so you can't feel, see, smell, hear or taste anything ever again. Forever.
So i could see EDI being allowed to continue operating on the Normandy. She could even lay claim to ownership.
The Normandy is pretty much what provides a "body" to the mind that is EDI. Its her senses, her means of movement and even her means to communicate with others.
A Court might come to the conclusion that it might be able to separate this Mind from this Body, but doesn't have the right to do so.
If EDI is found to be a sentient being and if she may claim ownership to the normandy, the Normandy and EDI will be treated as Prisoners of War. Having past affiliations with Cerberus it would be determined if this affiliation really is "past" and many People will have many Questions about cerberus.