This paper investigates a two-user downlink system for integrated sensing and communication (ISAC) in which the two users deploy a fluid antenna system (FAS) and adopt the nonorthogonal multiple access (NOMA) strategy. Specifically, the integrated sensing and backscatter communication (ISABC) model is considered, where a dual-functional base station (BS) serves to communicate the two users and sense a tag's surrounding. In contrast to conventional ISAC, the backscattering tag reflects the signals transmitted by the BS to the NOMA users and enhances their communication performance. Furthermore, the BS extracts environmental information from the same backscatter signal in the sensing stage. Firstly, we derive closed-form expressions for both the cumulative distribution function (CDF) and probability density function (PDF) of the equivalent channel at the users utilizing the moment matching method and the Gaussian copula. Then in the communication stage, we obtain closed-form expressions for both the outage probability and for the corresponding asymptotic expressions in the high signal-to-noise ratio (SNR) regime. Moreover, using numerical integration techniques such as the Gauss-Laguerre quadrature (GLQ), we have series-form expressions for the user ergodic communication rates (ECRs). In addition, we get a closed-form expression for the ergodic sensing rate (ESR) using the Cramer-Rao lower bound (CRLB). Finally, the accuracy of our analytical results is validated numerically, and we confirm the superiority of employing FAS over traditional fixed-position antenna systems in both ISAC and ISABC.