Not really? Depends on the type of Chinese Singaporean, most of the ancestors of the Nyonyas left during the Qing dynasty, and integrated into to the local cultures of Malaya. Those whose ancestors came and settled down in Singapore during 1819-1937 were mostly fleeing the decline of the Qing dynasty and the Warlord Era, and their descendants mostly identify as Singaporean first. The New Immigrants that came into Singapore post WW2 and after independence would identify more as Chinese first, mostly due to the more nationalistic education they received from the mainland.

Though I would like make a side point that being Singaporean isn’t tied to race, there are also large minorities of Indian and Malay Singaporeans that have also influenced each others’ culture.