I was playing as a longtooth strider druid and I wild shape (I forgot the exact animal) and I said I wanted to add my shift from my racial ability onto it, my dm said that isn't allowed but yet there is no rule stating you can't do that. So what does the official wizard of the coast rules say about something like that?

Also please note this dm is not a seasoned veteran, it's something they co on there past time.

Wild Shape

You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so.

If your DM decides that a form other than a shifter's isn't physically capable of shifting, I wouldn't question it.