Modelling multivariate survival data is complicated by the complex association structure among the responses. To balance model flexibility and interpretability, we propose a semiparametric copula model to modulate multivariate survival data, with the marginal distributions of the response components described by semiparametric linear transformation models. To conduct inference about the model parameters, we develop a two-stage maximum likelihood method and a three-stage pseudo-likelihood estimation procedure. We investigate the impact of model misspecification on the estimation of covariate effects and identify a scenario in which consistent estimation of the marginal parameters is retained even when the copula model is misspecified. The proposed methods are justified both theoretically and empirically. An application to a real dataset is provided to demonstrate the utility of the proposed method.