rt: A siddur has Shem Hashem. It’s as much Kedushah as any other Sifrei Kodesh. Why wouldn’t a Siddur be Tashmishei Kedushah?
And dafyomi, you are incorrect. The Tefillin boxes and straps both have a Din of D’varim Shel K’dushah (not even Tashmishei, they’re Kedushah Atzmah), not Tashmishei Mitzvah.