The emergence of entangled urban networks is a salient feature of globalisation processes. Understanding the drivers behind the growth of such networks, and in particular urban firm networks, is crucial for the economic resilience of urban systems. We introduce in this paper a generative network model for firm networks at the urban area level, which includes several complementary processes which are the economic size of urban areas at origin and destination, industrial sector proximity between firms, the strength of links from the past as well as the geographical and socio-cultural distance. An empirical network analysis on European firm ownership data confirms a role for each factor. We then simulate network growth for synthetic systems of cities, unveiling stylised facts such as a transition from a local to a global regime or a maximal integration achieved at an intermediate interaction range. We finally calibrate the model on the European network, outperforming statistical models and showing a strong role of path-dependency. Potential applications of the model include the study of mitigation policies for exogenous shocks such as economic crisis.