It self this act is not haram if it’s not done in such a way where Na Mahram involved also if it’s not making layer which is barrier for reaching water to ones skin then as for as Wudhu and Ghusl concern it would be in walid. But if it’s ink goes deep inside the skin and not making and membrane or layer then it’s allowed.
Now if we see the issue of tattoos in the light of ethics so we are not allowed to do any act according to Islamic Shariya which can be against Human Dignity and modesty. Because it act 99% done just for show off and if this is the intention then of course it would have problem and hard to say it’s okay. Allah (swt) has given a high rank to Human and our body is private and Amanat of Allah (swt) we should not use it to just portray it.
وَلَقَدْ كَرَّمْنَا بَنِي آدَمَ وَحَمَلْنَاهُمْ فِي الْبَرِّ وَالْبَحْرِ وَرَزَقْنَاهُم مِّنَ الطَّيِّبَاتِ وَفَضَّلْنَاهُمْ عَلَىٰ كَثِيرٍ مِّمَّنْ خَلَقْنَا تَفْضِيلًا
And surely We have honored the children of Adam, and We carry them in the land and the sea, and We have given them of the good things, and We have made them to excel by an appropriate excellence over most of those whom We have created.(17:70)