Above, you said "the closest cousin he has was 2nd cousin". Was this the one you are now saying shares 87 cMs, and is listed as 2nd-4th Cousin? On FTDNA, range of 2nd-4th Cousin means Suggested 3rd Cousin, not 2nd cousin. So this suggests there is no suggested 2nd cousin on FTDNA.
Do I understnad correctly - the same person matches you Dad with 87 cMs shared, longest block 26 cM - and matches you with 36 cMs shared, longest block 8? Have you looked on Chromosome Browser to see if your matches to him are in the same places as your Dad's? (so you know that yes, your match to him is thru your Dad - as opposed to a coincidence)
Is that match living in the US, or? Is he willing to share with you the names of his parents and/or grandparents? [grandparents should feel like less of a privacy invasion - and with any luck will be born long enough ago that they will appear in Records that can be found]
I think of "lots of Cohens" as suggesting Jewish ethnicity included to some extent, but if 87 is the highest shared cMs, then maybe not.
Approximately how many matches does your Dad have on FTDNA?
Have you posted a request in FTDNA's "Adoptees Forum" for suggestions on how to proceed? (That is the FTDNA link I gave above)
I am not good at all at manipulating Excel, so I cannot really explain how to use the Excel file you can get if Download all your results on Chromosome Browser to find the folks who match on the same segments. But basically, one can do something like that, then try to find the Family Trees of those folks who each match you in the same place, either thru their sharing or finding trees for them/their parents or grandparents on Ancestry or etc or doing the research your self to create a tree - then find mutual ancestor(s) on those trees - then chase and convince descendants further down the line to test (most folks trying to do this offer to pay for their FamilyFinder Test) to see if they match you any closer, and etc. Somebody there can probably explain it much more clearly, or give you a reference which will.
Do you have any info on your Dad's mother and/or know anyone who is a relative of his mother?
You say you have uploaded his results to MyHeritage and to GEDmatch - but No Matches - really, absolutely zero matches at both? Did you upload the results from FTDNA or from Ancestry? If it is truly zero matches, I would suggest try the other, since it is free, just to be sure same results.