'thrayI sAnkhyam yOga: pashupathi matham vaishnavam ithi'
'thrayI' refers to vEdha matham. Then sAnkhyam and yOgam. This pashupathi matham is the same pAsupatha matham that denigrates other deities and claims that Siva is the supreme lord. Similary, vaishnavam is the one which refutes (nirAkaranam) other deities and establishes the supremacy of Vishnu. VyAsar referred this only as pAncharAthram.
Both VyAsar and Pushpadanthar had accepted all these five schools. They have spoken in praise of them by equating them to five different rivers which finally take to the same sea. They have accepted even those who claimed that 'Only our deity is supreme'. However, as this attitude was considered to be against the true spirit of vEdha matham, they had chosen to classify these schools as those differing / deviating, completely from it [*vEda*].
In the case of Pushpadanthar, it is quite interesting. Because, he himself was a great sAmbhavA. The theme of his sthOthrA is nothing but Siva MahimA. He had even mentioned that these five schools are five different rivers leading to the confluence into the same sea of ParamEshvarA. Inspite of all these, he hadn't considered the saivaite pashupathi matham to be our sanAdhana dharmam, known also as vEdha matham. VEdha matham is the one that rejects nothing. As it is the one which absorbs everything, he had differentiated it from pAshupatham.
The third one is a poet. If there is significance to the utterance of VyasA, an important personality for all the three [*vEdAntic*] schools; if there is importance attached to the differtiation of the school which holds only Siva as the supreme from vEdA, by the saivaite Pushpadanthar himself; then there exists a great amount of value to the outpouring of a poet (kavi-vAkku) who remains totally unbiased without aligning himself with any particular school, and claiming neither superiority nor inferiority of any deity.
Let us look at the opinion of such a poet here. Who is that poet? It is the one who composed the story of Nalan as 'naishadham' - srI Harshar. When he was attempting to give a similie in a scene with extra ordinary literary merit, he also touches upon this topic of five schools.
The svayamvaram of Dhamayanthi is going on. She is in love with only Nalan. She is very particular that she has to garland Nalan alone, who is present in the svayamvaram hall. Even the four dEvAs - IndrA, VarunA, Agni and yamA - are pining for her. It is their desire to have her as their wife. They are also aware of her love for Nalan. Therefore, all of them disguised themselves exactly like Nalan and arrived there. Five Nalans are seated in the hall where the svayamvaram is going on. How could Dhamayanthi find the real Nalan [*from these five*]?
It is only here, srI Harshar says that similar to the presence of 'true' [*school of*] advaitha in the midst of five mathams, the presence of true Nalan amidst five Nalans caused confusion [*in her*] and describes the astonishment of Dhamayanthi due to her inability to identify him.
"panchama kOti mAthrE ... mathAnAm advaitha thathva iva sathya tharEpi lOka:"
'tharam' means the comparative degree. The poet hasn't said the other four schools are invalid/wrong, per se. Similar to VyAsar and Pushpadanthar, he also accepts them to be true and valid. But still, in comparison with them, as advaitA alone is found to be superior in expounding the truth exactly, he appreciatively referred to it as 'sathya thara api advaita'.
Similar to the way our AchAryAl has classified the sathyam into three types, Sri Harshar also categorised it as 'ordinary truth' and 'extra-ordinary truth' (uthama sathyam) and declared that out of the five schools mentioned in the [*MahA*]bhAratham, the other four are ordinary truth while advaita alone is the extra-ordinary truth.
In the bhAratham, in BhIshma vachanam, the term used is not 'advaitham'. The actual word employed is only 'vEdhA:'. But, whenever a school had set in the process of politicization such as 'Vishnu is the only Godhead' or 'Siva alone is the supreme lord', it came to be regarded, then, as differing from [*the true import of*] vEdic thought. The same view is echoed by the words of Pushpadanthar. Even in his composition, the vEdha matham has been referred to only as 'thrayI' and not by the name 'advaitham'.
In the post AchAryAl period, with the advent of saivaite and vaishnavaite schools, which invariably denigrate one [*deity*] and affirm the superiority of another [*deity*], it is only the smArtha matham of advaitins that had turned out to be the complete (pUrna) vEdha matham, accepting every thing. That is why, Sri Harshar had employed the term 'advaitham' itself [*in his verse*], in lieu of 'vEdha matham'.