Foundation models (FMs) have demonstrated strong transferability across medical imaging tasks, yet their clinical utility depends critically on how pretrained representations are adapted to domain-specific data, supervision regimes, and deployment constraints. Prior surveys primarily emphasize architectural advances and application coverage, while the mechanisms of adaptation and their implications for robustness, calibration, and regulatory feasibility remain insufficiently structured. This review introduces a strategy-centric framework for FM adaptation in medical image analysis (MIA). We conceptualize adaptation as a post-pretraining intervention and organize existing approaches into five mechanisms: parameter-, representation-, objective-, data-centric, and architectural/sequence-level adaptation. For each mechanism, we analyze trade-offs in adaptation depth, label efficiency, domain robustness, computational cost, auditability, and regulatory burden. We synthesize evidence across classification, segmentation, and detection tasks, highlighting how adaptation strategies influence clinically relevant failure modes rather than only aggregate benchmark performance. Finally, we examine how adaptation choices interact with validation protocols, calibration stability, multi-institutional deployment, and regulatory oversight. By reframing adaptation as a process of controlled representational change under clinical constraints, this review provides practical guidance for designing FM-based systems that are robust, auditable, and compatible with clinical deployment.