Good question LD. To really get an understanding of what the bible says you must understand the context. who is being written to and what are the circumstances that involve the (in this case) practice.

tatooing had a very different symbolism in those and later times. and they are incitefull of some questionable practices or ideals even today. there certainly are innocent people who tatoo for the sake of being in vogue, but their are tats that go beyond this.

Lev. 19 spoke to certain practices that was attached to idolatry, which jehovah hated (having another god before Him).

is tatooing a sin, nope. but idolatry so the symbolism is what is the danger.

here is what one commentary said about Lev 19:28 -Notes for Verse 28
Verse 28. Ye shall not make any cuttings in your flesh for the dead -- The practice of making deep gashes on the face and arms and legs, in time of bereavement, was universal among the heathen, and it was deemed a becoming mark of respect for the dead, as well as a sort of propitiatory offering to the deities who presided over death and the grave. The Jews learned this custom in Egypt, and though weaned from it, relapsed in a later and degenerate age into this old superstition (Isa 15:2 Jer 16:6 Jer 41:5).

nor print any marks upon you -- by tattooing, imprinting figures of flowers, leaves, stars, and other fanciful devices on various parts of their person. The impression was made sometimes by means of a hot iron, sometimes by ink or paint, as is done by the Arab females of the present day and the different castes of the Hindus. It it probable that a strong propensity to adopt such marks in honor of some idol gave occasion to the prohibition in this verse; and they were wisely forbidden, for they were signs of apostasy; and, when once made, they were insuperable obstacles to a return. (See allusions to the practice, Isa 44:5 Rev. 13:17 Rev. 14:1).
Notes for Verse 30
—Jamieson-Fausset-Brown Bible Commentary