Hey :)
Well, I have tattoos and I'm also a psychologist. Truth be told, we have an entire chapter to check about how our client looks like, and this of course, includes tattoos. This looks "chapter" is NOT as important as the others, but it is a source of information about our client - and any information we can get is helpful to better understand the client.
Some tattoos have meaning, some don't. We are interested in those with a meaning for the client.
Imagine a woman coming to your office, and she has a portrait tattoo in memory of her dead son. Maybe that's the reason she's there (the death of her son) and you'll want & need to find out more about it, and even if it's not, nonetheless it was an event that shaped your client one way or another.
And what if tattoos could talk? What story would they say? These are very good questions to ask the client once you have established a relation with him. Some people really like to talk about their tattoos!
There's also could be the case when a certain tattoo is related to a gang, to a rite of passage or even has a medical purpose (a tattoo which warns others in case of emergency that you have diabetes or certain allergies etc.) and it's important to find out the symbolism of these because it helps us, as I already said, to better understand the client.
I could go on and on, but the basic idea is that there's nothing wrong with your psychologist wanting to know more about your tattoos :)
P.S. - English is not my native language, so sorry for my mistakes.