A few thoughts to consider. First off, I am very well aware that you can't apply logistical thinking to its fullest extent when talking about a fantasy world. Let that preface apply to my reply. However, for the theater of the mind in trying to imagine how this ability comes to fruition would cause me to question that ruling on the matter. It's easy to imagine someone casting a fireball for example, even though it isn't real. But to have a short sword suddenly be able to strike a target 15 feet away without imagining some weird computer banding artifact effect or imagining Gumby in battle, because of a tattoo, is just hard to play out in my mind.
But the ability does offer some flavor to help us imagine and also, I think, help relate how this is to work. It says 'as inky tendrils launch towards the target'. So, what launches towards the target allowing it to be hit from 15 ft. away? Answer: The tendrils.
The reason it has you roll a melee attack is so that you know what to roll to try and hit. Its not a melee spell attack, its not casting a spell, its not a DC save. It qualifies as a melee attack for the mechanics of the tendrils hitting the target.
All that being said, to me, it sounds like if you are faced with a target 15 ft. away, you can make a melee attack roll, using your weapon modifiers to try and hit that target with your tendrils. Furthermore, it being classified as a melee attack would allow you to still make another attack if you have the extra attack feature. Therefore, it gets around the limitations of cantrips like Green Flame Blade that result in you losing your extra attack.
When it says 'In addition, your melee attacks deal an extra 1d6 force damage on a hit.' means that if the target is 5 ft. away from you, you can still apply the tendril damage to that target as well. Meaning the tendril, being capable of a ranged attack, is not hampered due to the target being beside you.
Another thought to consider. I believe what I said to be on track because this ability doesn't apply to a weapon such as a halberd. If the ability were to cause your weapon to 'stretch' to hit its target, then a reach weapon such as a halberd should also be able to 'Gumbify' to reach an additional 15 ft. But it doesn't. Therefore we can know that the weapon is not modified, your arms a not made of taffy to stretch and there is no phantasm spectral reach to it. So where do the inky tendrils come from to make this lengthy attack? I would propose they come from the tattoo. The tendrils manifest and lash out at any target you attack to do bonus damage. Or, you could make the attack motion at a target out of range to cause these tendrils to reach out and attempt to strike that target. That being said, I would even go as far to say these tendrils would work with the Sentinel feat. It says as one of its bullet points 'Creatures provoke opportunity attacks from you even if they take the disengage action before leaving your reach.' The tendrils, to me would qualify for this. So I think this tattoo still has plenty of uses if it was ruled to not have the melee weapon damage applied to the target outside the physical reach of the weapon.
Those are just my thoughts for consideration. After playing DnD 5e, it seems to me that most the time (not all), if a thing sounds to good to be true, it probably is. This ability falls under that umbrella for me, personally.