I have seen the trend of firms sending an email after rejection stating that they would like to keep the candidates' profile for future references and considerations.
In this case, according to GDPR in Europe at least, one should be able to request the firm to exclude their profiles.
In some firm cases though, the database are centralized, it might require them to go through several loops, only to add your profile back when you applied to similar firm at diufferent offices. Their system will certainly take notes on previous submission to avoid gamification of the process. There's a data security policies in effect.
Hence, it really depends on the firm. There should be no harm in asking but the likelihood of firms actually deleting your profile and considering it from a blank state might varied.