I already got the axe to cut the logs, but even after equip it still appears the message that I need an axe.

Am I doing something wrong or is it a bug??

I think it's a bug, have you tried starting a new game? Take the axe from the table and without touching anything else, go towards a tree.

Now it works, I even downloaded again, but the thing is that when I got the axe I equipped immediately but apparently you're not supposed to equip it to use it, so when I didn't equipped the axe it let me chop the trees.

Thanks for answering me.