Using `MapIndexed`:

 f[sols_List] := 
 MapIndexed[Set @@ {Symbol@ 
 StringJoin@(ToString /@ {#1[[1]], First@#2}), #1[[2]]} &, sols, {2}]

 f[Solve[y^2 == 13 x + 17 && y == 193 x + 29, {x, y}]];

 {{x1, y1}, {x2, y2}}

 (*{{(-11181 - Sqrt[2242057])/74498, 1/386 (13 - Sqrt[2242057])}, 
 {(-11181 + Sqrt[2242057])/74498, 1/386 (13 + Sqrt[2242057])}}*)