It's coincidence.

Ilya: from Russian Илья (Ilya), borrowed from Greek Ἠλίας (Ēlías), borrowed from Hebrew אֱלִיָּה (Eliyáh). It's typically a male name, although in English I wouldn't be surprised if people named some women "Ilya" because it ends in -a. I'm not aware of any flower with this name, but I don't speak Russian, so take it with a grain of salt.

Note Christian names in Russian often follow this same "double borrowing", just like Christian names in English most of the time follow a Hebrew → Greek → Latin (> French) → English path instead.

Julia: feminine of Julius, from Latin Iulius. The etymology is disputed, but apparently it comes from Iouilius (lit. "Jupitery", i.e. associated with the god Jupiter). Either way it's older than Christianity, and actually native in Latin, not borrowed.

Also, it wasn't a personal name in Classical times. It was a gens name; roughly a surname. That Julius Caesar for example was from the gens Julia, and his personal name was Gaius.

For a better example of weird etymological cognates: Ivan vs. John vs. Giovanni. They all ultimately backtrack to Hebrew יוֹחָנָן (Yoḥānān).

And it wouldn’t be the first name I see that changes for some random reason. For example, to my knowledge, the male name Tiago comes from a long line of mispronunciations starting at Jacob/Jacobus.

The key change here was rebracketing, the same change that transformed English "a napron" into "an apron". To explain it further:

Names of saints in Galician, Portuguese and Spanish are preceded by "Santo" (M) or "Santa" (F), but:

• if the name starts with a vowel, that final vowel is often elided, specially from Galician and Portuguese.
• if the name starts with a consonant, and is masculine, the whole /to/ gets elided. This is spelled as "San" in Galician and Spanish. (Portuguese nowadays diphthongises it to "São", but it's the same deal.)

I'll give you some examples (from Portuguese for my own convenience):

1. André (Andrew) → Santo André (St. Andrew), often pronounced as if it was "Santandré". Masculine name starting with vowel, so you keep the /t/ but elide the /o/.
2. Pedro (Peter) → São Pedro (St. Peter). Masculine name, starting with consonant, so you elide the full /to/.
3. Ana (Ann) → Santa Ana or Santana (St. Ann). Feminine name starting with a vowel, so you keep the /t/ but elide the /a/.
4. Sara (Sarah) → Santa Sara. Feminine name starting with consonant, so that /ta/ is both spelled out and pronounced.

Now, back into "Tiago". You got Latin borrowing the Greek borrowing of the Hebrew name יַעֲקֹב (Ya'akob). In Latin it became Iacob, Iacobus, Iacomus (yup, it was a mess.) That "Iacob" form evolved naturally into "Iago" into those languages. (Spanish also spells it as Yago.) Right?

But he was a saint. So you got to add "santo" to his name. It falls into the first case there, since the name starts with vowel: Santo Iago ~ Santiago. But there are so many versions of the name that people lost track of what it was supposed to be, so they interpreted it as "san Tiago" instead of "santo Iago". Then you get people naming their children after saints, and the name Tiago pops up.