A fluid reconfigurable intelligent surface (fRIS)-aided integrated sensing and communication (ISAC) system is proposed to enhance multi-target sensing and multi-user communication. Unlike the conventional RIS, the fRIS employs movable elements with adjustable positions, offering additional spatial degrees of freedom. In this system, a joint optimization problem is formulated to minimize sensing beampattern mismatch and symbol estimation error. An algorithm based on alternating minimization is devised to handle the resultant non-convex problem, where the subproblems are solved via augmented Lagrangian method, quadratic programming, semidefinite relaxation, and majorization-minimization. A key challenge is that the element positions affect both incident and reflective channels, leading to the high-order composite objective functions. As a remedy, the high-order terms are transformed into linear and linear-difference forms by exploiting the structural characteristics of fRIS and the channels. Numerical results demonstrate the superiority of the proposed scheme over conventional RIS-aided ISAC and other benchmarks.