Just because something is personal data doesn't mean you're prohibited from using/sharing/processing that personal data. You just need an Art 6 GDPR legal basis.
Here, the copyright holder is offering you a contract: you're allowed to freely use that image, as long as you attribute them properly. Since this attribution is necessary for fulfilling the license condition, you have a legal basis per Art 6(1)(b):
(1) Processing shall lawful [if] (b) processing is necessary for the performance of a contract to which the data subject is party […]
Furthermore, many copyright laws recognize moral rights: the right of the creator to be recognized for their creative works. This implies an attribution requirement, regardless of what the license says. So this attribution could also be argued to be necessary for compliance with your legal obligations, which is a legal basis under Art 6(1)(c).
Similar arguments could be made that the copyright holder has given their implicit consent, or that you have a legitimate interest in demonstrating the provenance of this image.
So are there privacy implications with attribution? Yes, absolutely. This is personal data, so you need to process it in accordance with the GDPR. However, the GDPR will not prevent you from fulfilling your contractual or legal obligations.