According to stories given in Shiva Purana, there is one Lord Shiva who is the main GOD and then Lord Rudra, who is a manifestation of Lord Shiva born from eyebrows of Lord Bramha. The TV serials hardly show this. This Rudra form is actually the one who resides on Mount Kailash and is also the one who married Mata Parvati. But, in most of the stories the name Shiva, Shankar, Mahadev, Rudra, etc are used interchangeably for both of them. So, we cannot be sure that the one who drank the poison is Lord Shiva or Lord Rudra. If he is Lord Rudra then I think since he is born out of Lord Bramha's eyebrows therefore he is not immortal.
According to Vishnu Purana and Shrimad Bhagvatam also, Lord Rudra was created by Lord Bramha but I think in these books Lord Visnhu/Lord Krishna has been shown as the main GOD instead of Lord Shiva and don't think they talk about 2 forms of Lord Shiva.
Following is the text from Shrimad Bhagvatam that describes the birth of Lord Rudra.
There was much anger generated in the mind of Brahmā, which he tried to control and not express. Although he tried to curb his anger, it came out from between his eyebrows, and a child mixed blue and red was immediately generated. After his birth he began to cry: O destiny maker, teacher of the universe, kindly designate my name and place. The all-powerful Brahmā, who was born from the lotus flower, pacified the boy with gentle words, accepting his request, and said: Do not cry. I shall certainly do as you desire. Thereafter Brahmā said: O chief of the demigods, you shall be called by the name Rudra by all people because you have so anxiously cried. My dear boy, I have already selected the following places for your residence: the heart, the senses, the air of life, the sky, the air, the fire, the water, the earth, the sun, the moon and austerity.... My dear boy Rudra, you have eleven other names: Manyu, Manu, Mahinasa, Mahān, Śiva, Ṛtadhvaja, Ugraretā, Bhava, Kāla, Vāmadeva and Dhṛtavrata.
So, I believe Lord Rudra might not be immortal but definitely being a param yogi and one of the main Gods, he has control over his body and senses. He was still capable enough to hold/control the poison in his throat.