I personally wouldn’t recommend that based on my past experience doing so.
If you are going to only use a single dropdown in your app, it may be fine to make a reusable element for the desire to format it to look the way you want regardless of the browser being use. However, some major drawbacks will be your ‘custom dropdown’ will not be a dropdown and therefore wouldn’t be able to be accessed via tab button.
Also, if you intend to use a dropdown in several areas of your app (I use them in every form…so pretty much used extensively through all apps), you wouldn’t be able to reuse the reusable element since you will need to set up different data sources to display the selection choices. Also, on a mobile device the users device will not recognize it as a dropdown element and will not behave the way you would want it to on a mobile device.
My personal recommendation if you are dead set on overriding the browser default CSS settings for dropdown elements is to investigate the custom css you could apply to your app to ensure the dropdown looks the same across all browsers. Unfortunately I do not have experience doing that, and also not 100% sure it can be done, but ultimately, I most definitely wouldn’t create a reusable element.