$ awk -v t=34897 '/^\[/{f=($4=="T("t")")} f' file
 [12] 03/31/21 08:33:30.080851 T(34897) _DBG message y 1
 [12] 03/31/21 08:33:31.457612 T(34897) _DBG message y 2
 test message line 2
 test message line 3
 test message line 4
 [12] 03/31/21 08:33:32.56341 T(34897) _DBG message y 3
 [12] 03/31/21 08:33:33.78123 T(34897) _DBG message y 3
 test message line 2
 [12] 03/31/21 08:33:34.56712 T(34897) _DBG message y 4

The above sets a "found" flag `f` every time it sees a line starting with `[`. If the 4th field on that line is `T(<target value>)` then f is set to true, otherwise false. When `f` is true the current line is printed.