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Do the words Wave(English) Welle(German) Vague(French) have the same Etymology as Val(Serbo-Croatian,Slovenian),Vlna(Czech,Slovakian),BолнаVolna.

All these words mean the same thing-Wave. but I found it surprising, given their relative similarity, that they have a different Proto-Indo-European root.

Wave/Welle comes from the Inde-European *wegh where Val/Vlna comes from *wel

Can anyone explain why these word that mean the same thing and are so alike still do not have the same etymology?

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First note that German Welle and English wave aren't cognates. Grimm derive German Welle from a verb walen "to writhe, to wallow, to roll" that is traced to Indogermanic *vel (in their notation, identical to your *wel). There is another German word for "wave", Woge (< wage) that derives from Indogermanic *wegh and that is related to English wave. French vague is probably a loan from Germanic.

I don't know details about the etymology of val and vlna. Are they inherited from proto-Slavic or are they loan words from German?

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    The Slavic words are cognate with, not borrowed from, German Welle. French vague is not borrowed from (High) German, but from Old Norse, via the Vikings.
    – fdb
    Commented Mar 13, 2018 at 12:51
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These sorts of coincidences happen quite often in comparative linguistics, just by random chance. For a famous example, German haben (~English "have") and Latin habēre mean the same thing, but come from totally separate roots; haben is instead cognate with Latin capere "seize", via Grimm's Law.

The main similarity between all these words is the initial /v/, which goes back to PIE. It might be a coincidence that *wegh and *wel started with the same phoneme, or it might not (if one affected the other), but all the later similarities can be traced back to this.

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    haben and habere are from separate roots, but is it really random chance that their uses correspond so closely? That would imply an experiment with no contact.
    – Adam Bittlingmayer
    Commented Mar 13, 2018 at 19:23
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    Draw the dividing line between way, valley, weave, vague, weigh -- is the wiggle a coincidence?
    – amI
    Commented Mar 20, 2018 at 23:53

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