1. You're performing the redirection on the strace command, so the redirection will be performed by your interactive shell strace will not be able to see relevant system calls.
2. The dup2 system call is performed inside of the child before the command execution. By default, strace will not follow children of your commands, so you won't be able to see the trace of the child itself. You will need to add the -f flag to your strace to make sure it also follows the children.

Notice the open and dup2 system calls are performed inside pid 31770, the child of the sh -c... command, before actually executing test.sh.

1. You're performing the redirection on the strace command, so the redirection will be performed by your interactive shell, and strace will not be able to see relevant system calls.
2. The dup2 system call is performed inside of the child before the command execution. By default, strace will not follow children of your commands, so you won't be able to see the trace of the child itself. You will need to add the -f flag to your strace to make sure it also follows the children.

Notice the open and dup2 system calls are performed inside pid 31770, the child of the sh -c... command, before actually executing test.sh.

Duplicating the file descriptors in the parent instead of the child

It is possible (depending on the implementation of your shell) that the duplication will be performed on the parent level before performing the clone. In that case, the parent will need to save the original STDOUT and STDERR file descriptors, and restore them after the child terminates.

$ grep -E 'dup2\(|clone\(|execve\(|(open|write|close|openat)\(.*tmp.txt|F_DUPFD|exit_group' trace.log
1094  execve("/usr/bin/sh", ["sh", "-c", "./test.sh > tmp.txt 2>&1"], 0x7fffd0324118 /* 20 vars */) = 0
1094  openat(AT_FDCWD, "tmp.txt", O_WRONLY|O_CREAT|O_TRUNC, 0666) = 3</tmp/tmp.txt>
1094  fcntl(1</dev/pts/0>, F_DUPFD, 10) = 10</dev/pts/0>
1094  dup2(3</tmp/tmp.txt>, 1)          = 1</tmp/tmp.txt>
1094  close(3</tmp/tmp.txt>)            = 0
1094  fcntl(2</dev/pts/0>, F_DUPFD, 10) = 11</dev/pts/0>
1094  dup2(1</tmp/tmp.txt>, 2)          = 2</tmp/tmp.txt>
1094  clone(child_stack=NULL, flags=CLONE_CHILD_CLEARTID|CLONE_CHILD_SETTID|SIGCHLD, child_tidptr=0x7f844aa71690) = 1095
1095  execve("./test.sh", ["./test.sh"], 0x7f844aa9dc08 /* 20 vars */) = 0
1095  fcntl(3</tmp/test.sh>, F_DUPFD, 10) = 10</tmp/test.sh>
1095  write(1</tmp/tmp.txt>, "hello\n", 6) = 6
1095  write(2</tmp/tmp.txt>, "./test.sh: 3: ", 14) = 14
1095  write(2</tmp/tmp.txt>, "whatfuckis: not found", 21) = 21
1095  write(2</tmp/tmp.txt>, "\n", 1)   = 1
1095  exit_group(127)                   = ?
1094  dup2(10</dev/pts/0>, 1</tmp/tmp.txt>) = 1</dev/pts/0>
1094  dup2(11</dev/pts/0>, 2</tmp/tmp.txt>) = 2</dev/pts/0>
1094  exit_group(127) 

See how before running each dup2 at the beginning, the parent (pid 1094) duplicates file descriptors 1 and 2 into file descriptors 10 and 11 respectively (using the fcntl system call and the F_DUPFD command) .

Then after the child (pid 1095) completes, the parent restores file descriptors 10 and 11 (the original STDOUT and STDERR) back to file descriptors 1 and 2.