If, for example, I say that I would repeat Subhanallah 10000 times, and Hz. Muhammad (pbuh) didn't advice such thing, is it considered bidah, or should I claim that this ibadah is part of religion in order for it to be bidah?
Given the answer to this question and the definition of bid'ah therein, you would have to claim that there was something special about doing that dhikr 10,000 times or that that specific number has certain virtues, or doing it at a certain time that is not proven from the sunnah, or doing it at a certain place, etc. In general we are encouraged to do dhikr as much as possible, so by itself there is no issue.
And Allah knows better.
(This is a Shia view)
Ibadah is two type. Wajib (obligatory) and Mustahab (liked). the example you said is not obligatory and is it is done without any belief that it is a rule in Islam there is no problem and allowed. but if one say it and believe it is part of Islam and consider it as a rule in Islam then it is Bidah and Haram.
About Wajib Ibadah adding something to them is Bidah.
Also any Zikr has a special spiritual effect on soul of human and it is not recommended use a personal made number of Zikr. it is recommended the numbers based on Hadith only. each Zikr for soul is like a drug for body and one should know spiritual effect of Zikar to can recommend numbers for each Zikr. it is recommended use only Zekr advices by Hadith like: Tasbih of Bibi Fateme Zehra (a.s)