Multi-axis additive manufacturing has emerged as a key enabler for fabricating high-performance composite structures, particularly those reinforced with continuous carbon fibers, where mechanical properties strongly depend on spatially varying deposition directions and fiber alignment. Compared to conventional planar printing, multi-axis deposition allows material to be placed tangentially to stress fields and geometric features, significantly improving strength and stiffness. Despite this promise, existing computational pipelines typically rely on indirect measures of geometric properties and manufacturability constraints, decouple layer generation from toolpath planning, and depend on discrete representations. These limitations reduce optimization fidelity and prevent direct control and coupled co-adaptation of layers and fiber toolpaths, which is critical for composite manufacturing. We propose a fully differentiable implicit neural-field-based optimization framework for multi-axis manufacturing that enables unified control over layer geometry, toolpath geometry, and collision avoidance. Layers and toolpaths are represented as level sets of implicit scalar fields parameterized by sinusoidally activated neural networks, providing continuous access to field values and derivatives. This formulation allows collision avoidance to be enforced explicitly during optimization and supports global collision reasoning beyond local curvature heuristics. Toolpath-level geometric properties, including orientation, spacing, and curvature, are directly encoded as optimization objectives, enabling coupled co-optimization of layers and toolpaths. The framework is particularly well suited for continuous fiber composite manufacturing, where precise spatial control of fiber trajectories is essential for achieving superior mechanical performance. The same representation and pipeline apply across multiple fabrication modalities, including multi-axis additive manufacturing and multi-axis milling, offering a unified computational tool for digital fabrication workflows. The approach is validated through physical fabrication and experimental evaluation of multi-axis manufactured parts, and an analysis of key hyper-parameters is provided to enable predictable control aligned with manufacturing intent.