Closer to the opposite.
Short version: people aren't protesting war associations, they're protesting racist associations, and the Union Army was the anti-racist army. If anything the Jackets would benefit from the new awareness, but the Union Army team identity is not clearly expressed.
Long version: After Reconstruction there was a successful effort to bring the South back into the country, Jim Crow racism was accepted and the fight of the South was recast as a noble lost cause. To symbolize the new Jim Crow order, the Confederate flag was added and monuments to Confederate generals went up around the country - except for Confederate generals who repudiated racism, they got no monuments.
It was always political. Now the politics have shifted. A lot of old textbooks still say the war had to do with tariffs or some other states rights issue, but the historical consensus is developing that it was slavery. The primary documents are clear enough. This isn't a left vs right thing - I saw a
video recently from the right wing Prager University that had military historians arguing that the War was clearly about slavery.
The Union Army is getting new respect because of the political shift - and you see it in movies like The Free State of Jones. If anything, one would expect the Blue Jackets to be embraced by more and more people because of the growing awareness about Civil War history. The Union side is increasingly embraced by everyone but the far-right, not that many things in history are like that! Unfortunately, most people think a Blue Jacket is some kind of bug, and don't even notice the Ohio flag in the main uniform. Whatever jersey you prefer, it's the case that the Union army identity is muddled and not clearly expressed. I think the team is missing an opportunity to gain some fans.