How and when were verses purportedly containing Messianic prophecies identified?

Famously, verses that Christians identify as messianic prophecies can appear a bit... cherry-picked. For example, the use of certain Psalms and the apparent mistranslation in Isaiah 7.

However, I was looking over the verses identified by Judaism as messianic prophecy and they appear no less ambiguous.

For example, Ezekiel 38:16 was identified here as messianic prophecy despite no apparent indication that it's not just about Gog. Isaiah 11 and other references appear likewise. Furthermore, my understanding is that Isaiah, Jeremiah, and Ezekiel predate the idea of a coming messiah. Therefore (if I'm correct) the authors could not have intended them to be understood as such.

Hence, my question: how did these verses come to be identified as referring to a messiah? Were these already identified by the first century, CE? Or is it possible that Christians and Jews identified different prophecies because they did so simultaneously?

I hope this is all clear and concise. Thanks for any help!