Spectrum sharing backscatter communication systems are among the most prominent technologies for ultra-low power and spectrum efficient communications. In this paper, we propose an underlay spectrum sharing backscatter communication system, in which the secondary network is a backscatter communication system. We analyze the performance of the secondary network under a transmit power adaption strategy at the secondary transmitter, which guarantees that the interference caused by the secondary network to the primary receiver is below a predetermined threshold. We first derive a novel analytical expression for the cumulative distribution function (CDF) of the instantaneous signal-to-noise ratio of the secondary network. Capitalizing on the obtained CDF, we derive novel accurate approximate expressions for the ergodic capacity, effective capacity and average bit error rate. We further validate our theoretical analysis using extensive Monte Carlo simulations.