Hi there,
you could do something like this to disable the browser styles:
.woocommerce form .form-row select,
.woocommerce div.product form.cart .variations select {
-moz-appearance: none;
-webkit-appearance: none;
appearance: none;
background-image: url('data:image/svg+xml;charset=US-ASCII,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20width%3D%22292.4%22%20height%3D%22292.4%22%3E%3Cpath%20fill%3D%22%23007CB2%22%20d%3D%22M287%2069.4a17.6%2017.6%200%200%200-13-5.4H18.4c-5%200-9.3%201.8-12.9%205.4A17.6%2017.6%200%200%200%200%2082.2c0%205%201.8%209.3%205.4%2012.9l128%20127.9c3.6%203.6%207.8%205.4%2012.8%205.4s9.2-1.8%2012.8-5.4L287%2095c3.5-3.5%205.4-7.8%205.4-12.8%200-5-1.9-9.2-5.5-12.8z%22%2F%3E%3C%2Fsvg%3E'),
linear-gradient(to bottom, #ffffff 0%,#e5e5e5 100%);
background-repeat: no-repeat, repeat;
/* arrow icon position (1em from the right, 50% vertical) , then gradient position*/
background-position: right .7em top 50%, 0 0;
/* icon size, then gradient */
background-size: .65em auto, 100%;
}
The background image i got from here:
https://css-tricks.com/styling-a-select-like-its-2019/
which is a useful article to help style the select box.