I believe that it’s impossible to tell what type of fish swallowed Jonah but my response would be does it really matter. Whether the story of Jonah is literal or not does not detract from the spiritual significance of the story. If the account actually happened, yes, there would be a fantastic event to go along with the story but God would be seen as going to extremes at the expense of one of His children. However, if the story of Jonah is an allegory, the story is not diminished in any way. In fact, the story is enhanced since God would not be seen as actually putting someone through that harrowing experience. The same thing can be said of Job.
The most important factor in the story of Jonah is the concept of grace and mercy. In Jonah Chapter 2, Jonah prays a prayer incredibly similar to that which Solomon prayed in 2 Chronicles 6. In that prayer, Solomon is asking for mercy. He asks the Lord to be merciful when Israel sins in the future. Note verses 36-39:
36 If they sin against thee, (for there is no man which sinneth not,)
and thou be angry with them, and deliver them over before their
enemies, and they carry them away captives unto a land far off or
near; 37 Yet if they bethink themselves in the land whither they are
carried captive, and turn and pray unto thee in the land of their
captivity, saying, We have sinned, we have done amiss, and have dealt
wickedly; 38 If they return to thee with all their heart and with all
their soul in the land of their captivity, whither they have carried
them captives, and pray toward their land, which thou gavest unto
their fathers, and toward the city which thou hast chosen, and toward
the house which I have built for thy name: 39 Then hear thou from the
heavens, even from thy dwelling place, their prayer and their
supplications, and maintain their cause, and forgive thy people which
have sinned against thee.
In these verses, Solomon requests that when Israel sins to such a degree that God will carry them away from Israel to a foreign land, that if the people would then turn toward the land and toward the temple and asks once again for mercy, then God forgive their sin.
In the next chapter (2 Chronicles 7): God tells Solomon that He has heard his prayer of mercy and now has selected the temple as a house of sacrifice; not animal sacrifice but a sacrifice of prayer. Here is 2 Chronicles 7:
12 And the Lord appeared to Solomon by night, and said unto him, I
have heard thy prayer, and have chosen this place to myself for an
house of sacrifice. 13 If I shut up heaven that there be no rain, or
if I command the locusts to devour the land, or if I send pestilence
among my people; 14 If my people, which are called by my name, shall
humble themselves, and pray, and seek my face, and turn from their
wicked ways; then will I hear from heaven, and will forgive their sin,
and will heal their land. 15 Now mine eyes shall be open, and mine
ears attend unto the prayer that is made in this place;
Jonah is applying the same truth to himself. He is acknowledging his sin of disobedience of refusing to go to Ninevah and that he has now been carried off to a foreign place far from Israel. He acknowledges that God has made the temple a house of sacrifice, a sacrifice of prayer. Note Johan’s words in Jonah Chapter 2:
1Then Jonah prayed unto the Lord his God out of the fish's belly, 2
And said, I cried by reason of mine affliction unto the Lord, and he
heard me; out of the belly of hell cried I, and thou heardest my
voice. 3 For thou hadst cast me into the deep, in the midst of the
seas; and the floods compassed me about: all thy billows and thy waves
passed over me. 4 Then I said, I am cast out of thy sight; yet I will
look again toward thy holy temple. 5 The waters compassed me about,
even to the soul: the depth closed me round about, the weeds were
wrapped about my head. 6 I went down to the bottoms of the mountains;
the earth with her bars was about me for ever: yet hast thou brought
up my life from corruption, O Lord my God. 7 When my soul fainted
within me I remembered the Lord: and my prayer came in unto thee, into
thine holy temple. 8 They that observe lying vanities forsake their
own mercy. 9 But I will sacrifice unto thee with the voice of
thanksgiving; I will pay that that I have vowed. Salvation is of the
Lord. 10 And the Lord spake unto the fish, and it vomited out Jonah
upon the dry land.
When God heard Jonah's prayer of mercy which he prayed symbolically “looking toward” (as Solomon requested) the temple, God responds by having the fish release Jonah from his bondage. I believe that when you see the incredible link between the two events (Solomon and Johah) you get an even more impactful story than if an actual fish swallowed Jonah.