Physical object?
Something to do with their name?
Biological?
Related to being president and first lady?
Just want to note that I changed the question because Joe and Jill Biden have one together. The dead animal is currently the only one who has one.
Joe and Jill are (I think) the only couple with one.
reply to Mahaloth:
Physical object? No.
Something to do with their name? No.
Biological? No.
Related to being president and first lady?
Joe and Jill because of their roles.
Is Champ the deceased animal in question?
I’m confused, now. When the new questions says “the presdient”, does it mean Biden or whomever is president?
reply to dirtball:
Is Champ the deceased animal in question?
No.
Joe and Jill Biden have one because they are president and first lady.
Would the next president after Joe Biden have one?
Was the deceased animal a celebrity when it was alive?
Was the deceased animal male or female?
Would the next president after Joe Biden have one? Yes
Was the deceased animal a celebrity when it was alive? Yes
Was the deceased animal male or female? Yes
Male 
Would the next president after Joe Biden have one?
The next president and their spouse would have one together.
Will Joe & Jill continue to have one after leaving office?
If the next President were a bachelor/ widower/ otherwise single would he have one?
Did previous Presidents and their spouses have one?
Was the dead animal noted for sports? Acting?
If the next president after Biden was a woman, would she and her husband have one together?
If the next president was gay, would they and their same sex spouse have one together?
If the next president was single or widowed would he/she have one?
reply to Peter_Morris:
Will Joe & Jill continue to have one after leaving office? No.
If the next President were a bachelor/ widower/ otherwise single would he have one? Yes.
Did previous Presidents and their spouses have one?
Yes.
All of them? All since a particular date or event? Some and not others?
reply to Peter_Morris:
Was the dead animal noted for sports? Acting?
No to both
Yes to all.
Not sure.
Is the deceased animal a dog? A horse? Another mammal? A domesticated animal? A wild animal? A normally-wild animal kept in captivity?
Is this a physical object (edit: oops, that one was already answered. Sorry!)? A title? A special privilege or “pass”?
Did James Buchanan (or any other unmarried President) have one?
ETA: This is really two questions as Buchanan may not have had one based solely on timing.