str does effect dmg. Depending on the base dmg of a pet, the damage modification may not be particularly noticable. For example, the difference between min str and max str rune beetles is only 3 dmg a hit. This is actually significant in long fights; not as important for short fights.
STR does impact dragon dmg. In fact, it impacts their damage more than it impacts any other pet because of their high base damage and high variance in str.
Sometimes it is hard to notice the difference in damage because of target's varying resistances and the range in base dmg, but there IS a difference.