Multilingual large language models (LLMs) exhibit substantial performance disparities across languages, with low-resource languages often lagging far behind high-resource ones. Federated learning (FL) offers a privacy-preserving path for decentralized multilingual adaptation, but existing methods typically aggregate client updates without explicitly modeling structural differences across languages, inducing gradient interference, amplifying negative transfer, and further disadvantaging underrepresented languages. We introduce FedLing, a linguistic-structure-aware federated learning framework. FedLing performs typology-conditioned spectral routing of client LoRA updates, decomposing them into linguistically transferable consensus subspaces and language-specific divergence subspaces. Unlike posterior-only decomposition methods that require accumulated training history to estimate transferability, FedLing explicitly mixes a round-one energy prior with typology-conditioned empirical subspace alignment through a single coefficient ρ. A continuous typological isolation score further prevents negative transfer for languages without close federation neighbors, and a reciprocal fairness proxy adapts upload bandwidth toward underperforming languages. Experiments across 40 languages show that FedLing consistently improves multilingual adaptation: +7.9 average ROUGE-L and +11.6 ROUGE-L on low-resource languages over FedAvg, with 52% variance reduction and 35% upload savings. The advantage is largest in cold-start and data-scarce regimes, validating the combination of energy-based cold-start routing and typology-conditioned posterior alignment.