Generally, we can find many things about Vāhana-s of Dieties, like Garuḍa, Nandikeśwara, Mūśaka etc. While very less is known about Simha of Bhagwati.
Bhagwan has his own Daśākṣara Mantra and its seperate Prayoga-s which is directed towards destroying the effects of abhicāra kramas, Anyways, that's a seperate Topic.
His Physical Body
Here it is brief description of Bhagwan’s form:-
In the Head of Bhagwan Simha, resides Sadā Śiva
In the Forehead of Bhagwan Simha, resides Pārvatī
His eyes are Chandra and Surya
In His ears, resides all Direction(Digdevatā)
His cheeks are Kubera and Yamarāja
His lips, resides Sandhya Devatā
His teeths are Aṣta Vasu Devatā
On his Tongue resides Varuṇa Deva,
The part where neck meets his head, there resides 27 Nakṣatra Devatā
In each of his Neck Hairs, resides Devī Lakṣmī
In the Neck of Bhagwan simha, resides Mahā Viṣṇu
In Chest resides Bhagwati Saraswatī
In his Heart, resides Devi Gāyatri
On either sides, resides Aświni Kumars
His Back, Indra Deva
In the wrists- Bhagwan Kartikeya
And again Bhagwan Nṛsimha is there in all his Claws
His genital, is Mahā Meru
His Tail is Brahma Deva, and the Hairs behind his tail are all the other Devatas like bṛhaspati, marud, rudras etc.
His Physiological Body
Sākṣāt Kālāgnirudra is his creulty
His Hūṃkāra, is Sākṣāt Carcikā Devī
His Prāṇa Vayu is Sapta Mātṛkā, His Apāna Vayu is Pitṛ Gaṇas
His sweat is Sākṣāt Gaṅgā and all the Rivers
The Strength and Valour of Simha is Chatur Vedas
We all Meditate on such a majestic form of Bhagwan, May Ambā’s dearest Vāhan destroy all the enemies inside us and take us to Ambā,
श्रीसिंहोऽखिलदेवतामयवपुः देवीप्रियः पातु माम् ॥
He is so much dear to Devi, that After her two names i.e Śrīmātā, Śrīmahārājñī, which both of them are her own eternal Nature.
Mother’s nature i.e. Vātsalya Bhāva and Royality(Maṇidweep Sāmrājñī) these two nature at a time can only be seen in Bhagwati Lalita Maha Tripurasundari only and no other God or Goddess,
So because these two nature are unique to Amba, these two personal Natures are told first and Just after that The Mahima of Her Simha is told as ‘Śrīmat Simhāsaneśwari’
The references of the post is taken from Śrī Uḍḍāmareśwara Tantra
Śrī Mātre Namaḥ.