Personalized fairness in recommendations has been attracting increasing attention from researchers. The existing works often treat a fairness requirement, represented as a collection of sensitive attributes, as a hyper-parameter, and pursue extreme fairness by completely removing information of sensitive attributes from the learned fair embedding, which suffer from two challenges: huge training cost incurred by the explosion of attribute combinations, and the suboptimal trade-off between fairness and accuracy. In this paper, we propose a novel Adaptive Fair Representation Learning (AFRL) model, which achieves a real personalized fairness due to its advantage of training only one model to adaptively serve different fairness requirements during inference phase. Particularly, AFRL treats fairness requirements as inputs and can learn an attribute-specific embedding for each attribute from the unfair user embedding, which endows AFRL with the adaptability during inference phase to determine the non-sensitive attributes under the guidance of the user's unique fairness requirement. To achieve a better trade-off between fairness and accuracy in recommendations, AFRL conducts a novel Information Alignment to exactly preserve discriminative information of non-sensitive attributes and incorporate a debiased collaborative embedding into the fair embedding to capture attribute-independent collaborative signals, without loss of fairness. Finally, the extensive experiments conducted on real datasets together with the sound theoretical analysis demonstrate the superiority of AFRL.