I'm not going to speculate on whether or not your stepson has ADHD. I haven't met him, I am not a developmental psychologist, and even if I was it would be impossible to diagnose over the internet. However, I've got a son with ADHD, and so I can provide some information relevant to why your two households are perceiving the same child so differently.
It's extremely plausible that his symptoms would manifest in different ways at different homes. There are a lot of symptoms of ADHD that can be "treated" by modifying the environment, and you may have been doing this completely by accident. Probably the biggest difference between Dad's house and Mom's house is school and homework. You have him at times when there is no pressure to get homework done, when he isn't tired after a long day of trying to focus... when he's doing fun stuff.
There's a huge difference between my son on a Thursday evening and the same boy on a Sunday morning. After days of struggling to stay focused, sit in his desk, do his schoolwork, do his homework, go to extracurriculars... he is completely stressed, out of patience, and hates everything. When those pressures are gone and he chooses his own activities (reading, playing, sleeping in, whatever) he is wonderfully agreeable and easy to get along with. School rules and procedures are particularly tough for a child who has a hard time focusing and remembering, and may lead to cycles of getting in trouble for something he can't really prevent, feeling like he can never do anything right so why bother trying so hard, getting in trouble for something he can't really prevent, etc... When he's removed from that cycle even for a couple of days, he can relax and is less prone to a frustrated explosion of emotion (tantrums).
If you want to bring up a discussion about medicating him for ADHD, it will be particularly challenging in this situation. He's not like that for us is a poor starting point; there are fewer stressors and triggers for him outside of school and/or homework situations. Understanding and acknowledging that he may be a very different child in a different home (and at school) may be the first step towards your two households finding a way towards helping him focus when he needs to — whether that's medication that a doctor has prescribed, a different homework structure on weeknights, an individualized education plan (IEP) or its equivalent in whatever country you're in.
The rest of this Answer may or may not be information that you already have. If you've already got all the background research out of the way, good for you :) However, I wanted to include it to be complete.
An ADHD diagnosis is not generated by teacher or parent complaints. A pediatrician needs to prescribe medication. Typically a child is referred to a more thorough diagnostic protocol with a developmental psychologist. The evaluation process includes filling out forms (sometimes just by parents, sometimes also by teachers, depending on where the concern and referral process was initiated), as well as direct observation of and conversations with the child to gauge his behavior.
ADHD doesn't always, or only, mean "is hyperactive and throws tantrums." The following list of possible symptoms comes from the National Institute of Mental Health (NIMH) webpage about ADHD. And yeah, a number of these are just things that children do: daydreaming! forgetting! impatient! An ADHD diagnosis doesn't come from just one of these things showing up occasionally, it requires six or more symptoms within a particular category. Get some clarification on what ADHD subtype your son was classified as (predominantly hyperactive-impulsive, predominantly inattentive, or combined hyperactive-impulsive and inattentive) and you'll have a better understanding of the concerns that led to this point.
Children who have symptoms of inattention may:
Be easily distracted, miss details, forget things, and frequently switch - from one activity to another
Have difficulty focusing on one thing
Become bored with a task after only a few minutes, unless they are doing something enjoyable
Have difficulty focusing attention on organizing and completing a task or learning something new
Have trouble completing or turning in homework assignments, often losing things (e.g., pencils, toys, assignments) needed to complete
tasks or activities
Not seem to listen when spoken to
Daydream, become easily confused, and move slowly
Have difficulty processing information as quickly and accurately as others
Struggle to follow instructions.
Children who have symptoms of hyperactivity may:
Fidget and squirm in their seats
Talk nonstop
Dash around, touching or playing with anything and everything in sight
Have trouble sitting still during dinner, school, and story time
Be constantly in motion
Have difficulty doing quiet tasks or activities.
Children who have symptoms of impulsivity may:
Be very impatient
Blurt out inappropriate comments, show their emotions without restraint, and act without regard for consequences
Have difficulty waiting for things they want or waiting their turns in games
Often interrupt conversations or others' activities.
