Arcane Focus =/= Spellcasting Focus
First off, I'd like to differentiate an arcane focus from a spellcasting focus. The former is a type of spellcasting focus specific to wizards, sorcerers, and warlocks. There are other types of spellcasting foci and those are specific to other classes.
PHB, page 151.
Arcane Focus. An arcane focus is a special item - an orb, a crystal, a rod, a specially constructed staff, a wand-like length of wood, or some similar item - designed to channel the power of arcane spells. A sorcerer, warlock, or wizard can use such an item as a spellcasting focus, as described in Chapter 10.
Druidic Focus. A druidic focus might be a sprig of mistletoe or holly, a wand or scepter made of yew or another special wood, a staff drawn out of a living tree, or a totem object incorporating feathers, furs, bones, and teeth from sacred animals. A druid can use such an object as a spellcasting focus, as described in Chapter 10.
Holy Symbol. A holy symbol is a representation of a god or pantheon. It might be an amulet depicting a symbol representing a deity, the same symbol carefully engraved or inlaid as an emblem of a shield, or a tiny box holding a fragment or a sacred relic. A cleric or paladin can use a holy symbol as a spellcasting focus, as described in Chapter 10. To use the symbol this way, the caster must hold it in hand, wear it visibly, or bear it on a shield.
PHB, page 154.
Musical Instrument. A bard can use a musical instrument as a spellcasting focus. Each type of musical instrument requires a separate proficiency.
What does the RAW say about using a tattoo as a spellcasting focus?
If you truly need an arcane focus, then this depends on if you consider a tattoo as an item similar to orbs, crystals, rods, staffs, or wands.
I would personally say, RAW, it is debatable. On the one hand, no, because tattoos are nothing like orbs, crystals, rods, staffs, or wands. They are gem-like or staff-like, and none of them are engraved into anything. Furthermore, tattoos are not "items" in the same sense as orbs or wands are items.
On the other hand, orbs, crystals, rods, staffs, and wands are nothing alike. The idea is that it has to be a focus designed to channel the power of arcane spells. As long as the tattoo was designed with that in mind, then yes, it can be a focus.
If you need a druidic focus, no. The rules explicitly list all the items you can use as a druidic focus.
As a holy symbol, I would say yes. Note that as a holy symbol, if tattooed to their hand, then they cannot wear a glove on that hand because for it to work, it must be held (you can't do that for a tattoo), worn visibly (which a glove will cover up), or bear it on a shield.
No, it has to be an instrument.
It is not hard to turn the tattoo into a disadvantage
If you wanted to disable him by removing the spellcasting focus, you would have to cut off his hand. But that's not the only way to give him disadvantage. An arcane tattoo cannot be removed, so the guy wearing it must be you.
As DM, you can make the character instantly recognizable due to the weird tattoo. Also, if other casters get a glimpse of it, you can make it so they can now divine the whereabouts of your character any time they want.
PHB, page 203
Casting some spells require particular objects, specified in parentheses in the component entry. A character can use a component pouch or a spellcasting focus in place of the components for a specified spell. But if a cost is indicated for the component, a character must have that specific component before he or she can cast a spell.
If the spell states that a material component is consumed by the spell, the caster must provide this component for each casting of the spell.
A spellcaster must have a hand free to access these components, but it can be the same hand that he or she uses to perform somatic components.
They will not be able to use the tattooed hand as a spellcasting focus if that same hand is wielding a shield, weapon, or any other object, because it will not be considered a free hand. Therefore, it will not work any differently than a regular spellcasting focus.
The difference is they won't be able to lose it/it can't be stolen. At the same time, they will not be able to hide it. As a fun story, you can make it a plot point that everyone who sees your player's tattoos will be able to identify this character and immediately assume friendly/hostile reactions as they will be instantly identifiable.
I would say though, wizards, sorcerers, and warlocks do not have to keep the focus visible, so they can hide it under a glove. Clerics are the only class that have the restriction that their holy symbol must be visible.
Also, losing the spellcasting focus only disables spells with material components. Your caster can still cast spells with verbal and/or somatic-only components, so removing a caster's focus is not an effective way to disable their spellcasting. So if your main concern was that they would never be disarmed, keep in mind that they will not be totally disabled if they had a regular focus and lost it.
And, even with a spellcasting focus, you still can't cast a spell if the material component is consumed or has a gold value anyway, if you didn't have that material component at hand (and then you wouldn't need a focus for that spell).
It is not overpowered
All this to say, it is not that big of a deal. They gain something minor and extra -- their focus is bounded to their skin -- but it has its own minor and extra drawbacks. I would say they balance out.