The tamed deer normally can't attack enemies but somehow it attacks my character when I activate its song skill. It comes near the character and headbutts him :) for about 40-50 damage.
Before this started happening, my character missed a fireball and hit it accidentally, could this be the reason?
Even if I cancel the heal skill, it continues attacking...
After it is killed, bringing it back did not change this behaviour.
Screenshot: https://imgur.com/a/nkbYu7l

OK, I think I found the reason for this behaviour.
Let's say I have my melee/ranged attacks in quickslot bar 1.
I put my spells in quickslot bar 2.
I also put the deer's heal skill in quickslot bar 2. Then doing the following:
- I right click on deer's heal in bar 2,
- Before it triggers, I switch to quickslot bar 1 to use my ranged skills,
- When I end my turn using a skill in quickslot bar 1, the deer will either stay still without using its skill or attack my character in melee.
This also explains why my other pets sometimes did not use their skills even after I activate them. They were simply in the other quickslot bar.
That means for a pet's skill to work correctly, the quickslot bar they are in must be active at the end of your turn.
Good game marred by such annoying design problems.

Once the deer starts attacking you, it won't leave you alone until you manage to click on it and move it away. It killed my character once during a difficult battle :)