Whether you can apply to both usually depends on the firm, office, and eligibility rules for that recruiting cycle. Some systems may allow both applications, while others expect you to choose the path that matches your graduation date.
The bigger issue is fit. If you are clearly eligible for full-time recruiting, applying for an internship as well can create unnecessary confusion unless there is a specific reason the recruiter has advised you to do so.
The safest move is to check the posting requirements carefully and ask recruiting if the situation is genuinely ambiguous. A short clarification is better than submitting two applications and hoping the system interprets them correctly.
You want your application to tell a coherent story about where you are in school and what role you are ready for next. A useful reference is moving between consulting internship and full-time recruiting.