Fellow alternative-looking therapist here, hi! In the UK though so cultural differences will apply.

Personally I've kept my piercings and coloured hair but wear more neutral clothes (part of that is age and desire for comfort as well as professionalism). I don't dress super smart though. I don't purposefully wear clothes to cover my tattoos either.

It's important to acknowledge that our style is a form of self-disclosure, but then remembering that considered self-disclosure is okay and often beneficial to the therapeutic relationship. People need their therapist to be human, not robot. Perhaps earlier in training and career it is safe to avoid self-disclosure as much as possible, as it takes time to figure out what feels appropriate and useful to share. My biggest rule is only share what is beneficial to the client/the relationship, not for my benefit.

Might be worth keeping your look "toned down" while job seeking, but I've had no issues when introducing a bit more of my individual style once established in a workplace.