Assalamualaikum. Is taking an oath with a name other than Allah’s name considered as shirk?
Alhamdulillah, praise and gratitude to Allah SWT for His countless blessings for us all. Praise and salutations to our beloved Prophet Muhammad PBUH, his family, companions and all those who follow his footsteps until the Last Day.
Oath in Arabic is al-Aiman (الأيمان). It is the plural form of the word al-Yamin (اليمين). In principle, it means hand for to take an oath, the Arabs would usually raise their hands. 
Whereas, the term means to strengthen something that one is vowing with by presenting something that is specifically revered. This is done by using one of the letters ba’, waw or ta’ (in Arabic). It is by saying billahi, wallahi or tallahi which means ‘by Allah’. 
A vow should only be made by using the names or attributes of Allah SWT. In a hadith narrated from Abdullah bin Umar R.Anhuma, the Prophet PBUH said:
مَنْ كَانَ حَالِفًا فَلْيَحْلِفْ باللَّهِ، أَوْ لِيَصْمُتْ
“If anyone swears, he must swear by God, or keep silent.” 
The Ruling of Taking an Oath with Other Than Allah’s Name
In a narration from Ibn ‘Umar R.Anhumma, he once heard a man say: “No, by the Kaaba.” Ibn ‘Umar R.Anhuma then said: “Nothing is sworn by other than Allah, for I heard the Messenger of Allah (ﷺ) say:
مَنْ حَلَفَ بِغَيْرِ اللَّهِ فَقَدْ كَفَرَ أَوْ أَشْرَكَ
“Whoever swears by other than Allah, he has committed disbelief or shirk.” 
Al-Syaukani in Nail al-Autar  explained the word ‘disbelief’ or ‘shirk’ in the above hadith explains the severity in terms of the warning and rebuke in the matter of taking an oath.
The source of the restriction is that swearing with something leads to glorifying it. Glorify in truth only belongs to Allah SWT, its essence and attributes. Hence, fiqh scholars agreed that swearing with other than Allah SWT is invalid for the restriction proves the damage of the matter that is restricted.
Scholars have differing opinions, is taking an oath with other than Allah is haram or makruh?
Ibn Hazm [ruled] it haram (prohibited). While most scholars ruled it makruh. Actually, it calls for more details, if a person takes an oath with other than Allah SWT and believes in it the same as his belief is in Allah SWT, then such belief is disbelief.
However, we would like to emphasize that swearing with other than Allah’s name is restricted and absolutely prohibited. From Abu Hurairah RA, he said the Messenger PBUH said:
لاَ تَحْلِفُوا بِآبَائِكُمْ وَلاَ بِأُمَّهَاتِكُمْ وَلاَ بِالأَنْدَادِ وَلاَ تَحْلِفُوا إِلاَّ بِاللَّهِ وَلاَ تَحْلِفُوا بِاللَّهِ إِلاَّ وَأَنْتُمْ صَادِقُونَ
“Do not swear by your fathers, or by your mothers, or by rivals to Allah, and swear by Allah only, and swear by Allah only when you are speaking the truth.” 
Syeikh Syams al-Haq Azim Abadi explained: Don’t you swear with your fathers. The same with your children. Then surely, it is aula (prioritized) to not swear by idols.
In al-Nihayah, it is stated: The word ‘’ is the plural to ‘’ which is something that is similar or comparable in everything and contradicts it. The meaning here is that they are gods other than Allah.
Scholars have differing opinions on the prohibition of this issue. According to madhhab Maliki, there are two opinions. This is as stated by Ibn Daqiq al-‘Id and the famous opinion according to them is it is makruh. Likewise, it is also a khilaf in madhhab Hanbali. However, the most well-known according to them is that it is prohibited and this is the opinion held by al-Zahiri.
The same is stated in madhhab Syafie which states it is makruh as stated by Imam al-Haramain. However, if one revered and glorify with the matter he swears by, then it is haram. Moreover, such belief could lead to disbelief. 
Taking an oath with other than Allah SWT is a bid’ah that could lead to disbelief. The aspect that makes it bid’ah is taking an oath is a form of worship. If it is done not for the sake of Allah, it means we are including something into the religion something that has nothing to do with it. 
 See Taudhih al-Ahkam, 7/105
 See al-Qaul al-Mufid, 2/213 and Taudhih al-Ahkam, 7/105
 Narrated by al-Bukhari (3836) and Muslim (1646)
 Narrated by Abu Daud (3251), al-Tirmizi (1535) and Ahmad (6072)
 Narrated by Abu Daud (3248) and al-Nasa’ie (3769)
 See al-Bid’ah al-Munkarah, by Syeikh Dr Wahbah al-Zuhaili, pg. 27