There was a fairly effective push to dismantle groups linked to Trump's electoral success [0]. They had a similar air of bullshit charges that only became an issue because prosecutors wanted to achieve political ends.

It is weird, but it is likely he is being threatened. If they can get Trump arrested for this, they can certainly put Cohen in jeopardy.

[0] https://www.nytimes.com/interactive/2018/08/21/us/mueller-tr...

EDIT Also https://en.wikipedia.org/wiki/Steve_Bannon#2022_New_York_sta... is probably going to fall into the same basket

That wasn't the only crime he pled guilty to. Likely had to plea guilty to all crimes to get his deal.

Cohen was not crime lawyer. And any high-level lawyer will tell you that even them will not represent themselves in a criminal investigation nor are they immune from make false confessions under the pressure of the federal government.