The existence is not guanranteed of a strongly regular graph, $d$-regular, every pair of adjacent vertices having $a$ common neighbors, every pair of vertices not adjacent having $b$ common neighbors. But if it does exist for a set of parameters, should the graph ,up to isomorphism, be unique?
No. For instance, there are $15$ different Paulus graphs with parameters $(25, 12, 5, 6)$. For more information on strongly regular graphs, be sure to check Andries Brouwer's database and lecture notes.