Yes, the name of the Foreign Branch and the parent company must be identical. Since it is a representation of the parent company, the names have to be exactly identical. However, if there is a requirement for licenses or if the name of the company is similar to an existing entity, there will be a need for an appeal, which may lengthen the process of your branch registration.
Suggested Readings
Does a Singapore Branch Office need to appoint a Company Secretary?
How do I register my Branch Office with the Accounting And Corporate Regulatory Authority (ACRA)?
What is the procedure for setting up a Branch Office in Singapore? How long does it take?
What tax incentives will I enjoy if I were to register a subsidiary company rather than a foreign branch?