Is the city responsible for sewer backup?

Not automatically. A sewer utility generally maintains its public mains, while the property owner usually maintains the private connection, or lateral. The exact boundary varies locally. Responsibility for fixing a pipe is also different from legal responsibility to pay for damage inside a home. Report the backup to your sewer utility, ask where the obstruction occurred, and keep its incident number and findings. If you believe a public-system failure caused damage, ask the municipality about its claims process and deadlines and notify your insurer. A public-main backup does not by itself guarantee reimbursement.