Why is set -e also ignored for functions called via command substitution

I've seen questions about this topic, but they talk about constructs like if or ||. I don't understand why the same behavior seem to happen for substitution ($())?

$ my_function() {     echo "the following command could fail:";     false;     echo "this is after the command that fails"; }
$ (set -ex; var=$(my_function); echo "$var" )
++ my_function
++ echo 'the following command could fail:'
++ false
++ echo 'this is after the command that fails'
+ var='the following command could fail:
this is after the command that fails'
+ echo 'the following command could fail:
this is after the command that fails'
the following command could fail:
this is after the command that fails

vs

$ my_function() { set -e;     echo "the following command could fail:";     false;     echo "this is after the command that fails"; }
$ (set -ex; var=$(my_function); echo "$var" )
++ my_function
++ set -e
++ echo 'the following command could fail:'
++ false
+ var='the following command could fail:'

errexit seems to behave as expected for simple commands:

$ (set -ex; var=$(false); echo "$var" )
++ false
+ var=

If I use read instead I get the expected behaviour:

$ my_function() {     echo "the following command could fail:";     false;     echo "this is after the command that fails"; }
$ (set -exo pipefail; my_function | read var; echo "$var" )
+ my_function
+ echo 'the following command could fail:'
+ read var
+ false