thanks to Charles Duffy! I had to check the box "Builtin version of 'test'" in the shell section of the make menuconfig menu of busybox, to enable string comparison. Now the code works.
Since I had grep activated I first tried another solution which also worked but has a bad performance for sure:
#!/bin/sh
msg=$(dmesg | tail -n1)
echo "$msg"
tasklet="Tasklet grp12"
if ( echo "$msg" | grep "^$tasklet$" )
then
echo "Test was successful, Strings are equal."
else
echo "Test failed, Strings are not equal."
fi
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