I've always noted surface similarities between the account of Polycarp's martyrdom and Christ's crucifixion as recorded in the gospels.

But I haven't seen the similarity to the same degree that Lightfoot describes (Lightfoot, Ignatius & Polycarp, vol II pt 1, pp. 594-595). He writes:

The writers [of Martyrdom of Polycarp] betray an eagerness to find parallels between the sufferings of their martyred bishop and the passion of Our Lord. ... Accordingly the idea of literal conformity to the sufferings and death of Christ runs like a thread through the whole document. Some of the coincidences are fairly obvious; in other cases the parallelism is highly artificial.

The list is a long one.

• The officer who apprehended Polycarp was named Herod (MPoly 1).
• Polycarp's location betrayed by a slave boy is likened to Judas' betrayal of Jesus, though this is a bit of a stretch. (MPoly 2)
• Polycarp, like Christ, was betrayed by those in his own "household' (though again, a bit of a stretch). (MPoly 6)
• Polycarp predicted his death by fire, much like Christ prophesied his own death (MPoly 5)
• Polycarp didn't flee to escape his capture, much like Christ did not hide or run away (MPoly 1)
• Polycarp's pursuers were armed sought him "as a robber" (MPoly 7)
• Polycarp's declaration of "God's will be done" (MPoly 7, cf. Mt 26.42)
• A voice from heaven encouraging Polycarp (MPoly 9, cf. Jn 12.28)
• Polycarp's body pierced (MPoly 16, cf. Jn 19.34)
• Interference of Jews in disposing the body (MPoly 17, cf. Mt 28.62)

Lightfoot goes into more detail at several points and adds other points to his list that I've not included above. Most interesting to me is the thought that several martyrdom stories, not just Polycarp's, contain elements that seem to be related back to the suffering and crucifixion of Christ.

Apparently some use this as evidence that these sorts of stories are false or are embellished to the point so as to not be reliable. But this sort of conclusion seems invalid to me (as it does to Lightfoot). Why wouldn't Christians pick out and emphasize supposed similarities between the suffering of a martyr and the crucifixion of Christ? The point isn't that Polycarp was some sort of Christ figure -- far from it. The point in the narrative of Polycarp's death is that he died a death worthy of a saint. There may very easily be some embellishment in the account (the bit about all the blood extinguishing the fire, a dove flying out of Polycarp's body, and the sweet resultant aroma seems a little much to me) but that doesn't mean we throw the whole thing out as fiction.