If you have entered your VAT number under 'company details' in your profile, the company name / trade name and relevant info is automatically loaded from an external (European) VAT database. This company info is collected by them from the national tax authorities. So this is the info that will then be shown in your account. Unfortunately, that cannot be modified within Gearbooker. Should your company be registered with a main company name and other underlying names, then it's only possible to show the main name in your Gearbooker profile because this is the only name that it's being retrieved from the VAT database.
If you have changed your company name or address at the tax authorities, this will be synchronized with the European database every so often and Gearbooker will automatically retrieve it from there. You can also manually click on the 'validate and save' button on this link.
You can choose not to show the company name, but just your first name, via https://www.gearbooker.com/manage/updatevat by setting the corresponding button to "no".