[mseebach]

OP isn't perfectly unambiguous on the subject, but it's an industry conference, and since asking the manager is a thing, I think it's fair to assume that it's a work conference and that the manager actually has a legitimate say in this - participants in industry conferences are typically identified as representing their company.

[turc1656]

Exactly my thoughts. If this to be done during normal work hours when this person would normally be in the office and working (almost certainly the case), then the manager would definitely need to approve this, especially since there is usually a cost associated with attending. If the employee is willing to pay themselves, that still leaves the issue of that time being utilized for non-approved activity during work hours. If the employee decides to take that day off and use their personal time they can certainly attend on their own dime, but the employer can still stipulate that they are explicitly not to represent the company at that event and not introduce themselves as an employee of the company. I don't know what the employer's reasons would be for this, but if that was their desire they are certainly allowed to choose who represents them and when.

[mseebach]

Exactly. And I think it's safe to stop after the first statement. OP made no mention of offering to go on their own time and dime, and being rejected.