Q: Is saying “Allah Allah” ﷻ a bid’a?
A: No, it is not a bid’a.
1: Different Tools of Calling
There are many tools of calling in Arabic, آلة النداء. Ya يا is only one of several, not the only one.
The alif hamza is one of those أ. If you say a’Zayd أزيد you’re calling upon Zayd.
With istighatha استغاثة or calling for help, you don’t have to exclusively use يا. You can just say “Allah (ءالله)!” Though every example in the Qur’an and hadith of the hamza with the Divine name is used for a question (to my knowledge), as the hamza can also be used for asking a question. IE:
آللَّهُ خَيْرٌ أَمَّا يُشْرِكُونَ – 27:59 – “Is Allah better or what they associate with Him?”
(note the آ alif with madda indicates a merged hamza and alif: ءا)
2: Implicit Harf Nida’
You can also call with an implicit harf nida’ (آلةُ نِداءٍ مُقَدَّرةٌ). One example is in the Qur’an 12:29:
يُوسُفُ أَعْرِضْ عَن هَذَا – “Yusuf, turn away from this.”
al-‘Irab al-Maysir states: يوسف منادى محذوف منه حرف النداء – “The word ‘Yusuf’ is called upon with the harf al-nida’ omitted from it.”
Every other ‘irab I checked says the same.
Additionally, dhikr simply means remembrance of Allah ﷻ in general, and when you say His name you’re remembering Him.
There is also a marfu’ sahih Hadith:
“ لاَ تَقُومُ السَّاعَةُ حَتَّى لاَ يُقَالَ فِي الأَرْضِ اللَّهُ اللَّهُ ”
“The Hour will not be established until ‘Allah, Allah’ is not said on the earth.”