You are conflating two different scripture passages.

Luke’s gospel describes the conception by saying “the Holy Spirit shall come upon you and the power of the Most High overshadow you.” No mention if animals.

At Jesus’ baptism, the Holy Spirit descends upon him in dove form and a voice from heaven speaks. So while the Spirit is pictured as a dove, this imagery does not appear in biblical accounts of Jesus’ conception.

As to whether the story owes any inspiration to Grecian mythology, there’s no good way to know. However, we do know that both The account of the annunciation and the book of Acts were written by Luke. Luke was a traveler with Paul later in his ministry. Midway throught the book of Acts Luke begins using the word “we”. It is unclear when Luke joins the party, but he seems to have been a Greek convert. So knowledge of Greek mythology would not be out of the question. In fact, Acts offers an account of Paul being mistaken for Zeus in one city. However, it seems unlikely for the conception story to be specifically influenced by Zeus’ animal transformations.

You're conflating Biblical accounts here. Neither Matt. 1:18-24 nor Luke 1:26-38 mention a dove.

The Matthew birth narrative mentions that Mary was impregnated by the Holy Spirit, but the focus in Matthew's is a bit more on Joseph's response to the pregnancy. In the Lucan birth narrative, Mary is visited by an angel and the focus there is on Mary's conversation with the angel Gabriel.

Where there is a dove is in the baptismal narratives: Mark 1:10, Matt. 3:16, Luke 3:22. These three Gospels are called the synoptic Gospels since they're so similar. The short version to explain this similarity is that the popular authorship hypothesis in Biblical scholarship says Mark was written first, both Matt. and Luke share Mark as a common source which they heavily draw from, but both also use other sources and might have also had other sources in common. So Mark and Matthew being largely identical and Luke having a slight variation is expected. John's account is slightly different, but also has the dove imagery.

In Mark the terminology is "the Spirit, like a dove, descending upon him" (New American Bible Revised Edition) and Matthew's wording is very similar. Luke has the phrase "the holy Spirit descended upon him in bodily form like a dove." In John's Gospel, John the Baptist is not said to have baptized Jesus but rather he tells an audience:

“I saw the Spirit come down like a dove from the sky and remain upon him. I did not know him, but the one who sent me to baptize with water told me, ‘On whomever you see the Spirit come down and remain, he is the one who will baptize with the holy Spirit.’ Now I have seen and testified that he is the Son of God.”

(John 1:32-34)

So the dove imagery is used to inaugurate Jesus' ministry and demonstrate that Jesus is the Messiah, but it's not an image associated in the canonical Gospels with Jesus' infancy narratives (keeping in mind that Mark is the synoptic without an infancy narrative and John opens with a bit more of an abstract meditation; but the two synoptics have radically different accounts). But even on the dove imagery, there's some slight disagreement. Luke is a bit of an outlier in saying that the Spirit literally appeared as a dove. That the Spirit appears as a dove there doesn't necessarily mean that the Spirit always appears as a dove, though. Usually the Spirit isn't assumed to take the form of a literal creature. Christians would come to identify the spirit of wisdom in Proverbs as being the Holy Spirit, but there the Spirit is personified as a woman calling to men in the streets (Prov. 1:20ff.).

On the dove imagery in particular, Leon Morris writes::

All four Evangelists mention the descent of the Spirit as a dove. Matthew and Mark tell us that Jesus saw this and John that the Baptist saw it. Each may have been speaking of a subjective vision, but Luke’s in bodily form shows that there was an objective reality. The symbolism is puzzling, as the dove was not an accepted symbol of the Holy Spirit (though a few late Jewish writings do use it). Rather the dove stood for Israel. However, there is no doubt that here it is a sign of the coming of the Holy Spirit. What happened at Jesus’ baptism led Christians to see the dove as a symbol of the Holy Spirit; they did not take the symbolism from Jewish or Hellenistic sources.

Leon Morris, Luke: An Introduction and Commentary, vol. 3, Tyndale New Testament Commentaries (Downers Grove, IL: InterVarsity Press, 1988), 117–118.

I'd be cautious about making so firm a claim (I wouldn't be surprised, personally, if there are non-Christian analogues though I personally do not know of any), but the importance here is that Luke is, again, the odd one out for being as firm as he is on this claim and it doesn't seem like this was a particularly common image for the Spirit before the baptism narrative in the surrounding culture.

Added: That being said, this is a good question and I think is important because it gets to something else -- how ubiquitous the dove is as a Christian symbol for the Spirit. In my own denomination, the Cross and the dove are extremely common to see in our churches and on denomination publications since due to the logo of one of our larger bodies. As an aside, I hate that logo because it took me an embarrassingly long time to realize one half of it is a dove. I don't know if it's a bad logo or if I'm just a bit more dense than I'd like to admit, but I just dislike that logo.

A lot of Christian imagery conflates different accounts into one coherent account (the typical Christmas pageant play is a harmonization of the infancy narratives from Matthew and Luke). I would not be surprised if there is, in fact, imagery out there that has the dove present when Mary is impregnated. We get so used to these images it's real hard to realize that the versions of the stories we're used to are not actually as they're written or that we bring in imagery from later narratives into earlier ones (which is what Christians do when they identify Wisdom in Proverbs with the person of the Trinity called the Holy Spirit -- that's not how Jews would understand that character, but it's impossible for those of us who are Christian to see Wisdom as anything but that person of the Trinity).