The running mate wouldn't be considered the candidate proper. The party would have to nominate and run someone through proper procedure. That doesn't mean the running mate couldn't be the candidate though. If Biden picks Kamala and he drops out, the party honchos could tap her to run in his place (and she would have to pick a running mate of her own).
'Course we're talking about the Democrats so anything is possible. "Rules"? There are no rules anymore.
'Course we're talking about the Democrats so anything is possible. "Rules"? There are no rules anymore. Exactly. Remember what the Democrats did in the New Jersey Senate race of Robert Torricelli.
Torricelli was involved in a major scandal, and was losing.
The Democrats replaced him with former Senator Lautenberg, long after the legal period to do so. They just did it, and their pet courts approved of it.