Often, optometrists will not write a generic prescription because they want to verify the fit of the specific lens they gave to you. Lenses with the same specs from different brands can fit differently and they don't want to be liable for any injury/damage resulting from a poorly fitted lens (badly fitting lenses can 100% fuck up your eyes even if they feel totally comfortable) so at least in my country they usually prescribe a specific lens.
Technically all you need to order contacts from any brand are the base curve, diameter, and lens power. Make sure that the lens you order has the same measurements as the one on your contacts prescription. Try to get one made from the same material too if possible. The lens material is usually written on the packaging.
Follow those gudielines and you have a much lower risk of causing problems (but still a risk, so just be aware). You can ask any optician to check the contacts while you're wearing them to make sure they fit properly if you're worried. Good luck anon!