Hanuman worshipped the Avatar Sri Rama and both worshipped the Shivalingam .
In Rameswaram, Rama installed the Shivalingam which is known as Ramalingam and a temple is built around it named as Ramanathaswamy. Hanuman brought a Shivalingam from the Himalayas and installed it as well. It is known as Vishwalingam.
The temple's name, Ramanathaswamy, an another name for Shiva, means Rama's Nath or Lord who is Shiva.
Rameshwaram in turn, also means Rama's Eashwar or God, that is Shiva.
Rama instructed that the Vishwalingam, the Lingam brought by Hanuman from the Himalayas be worshipped first, and this tradition continues even today.
The Shivalingam is considered in the Shivapuranas or Shaivite scriptures to represent a cosmic pillar of light.
The Prajapita Brahmakumaris
too state that the Shivalingam represents God as a point of light.
Interestingly, there are many interesting correlations between God and light in other major world religions as well. I have briefed on this in this thread