Because of my dietary restrictions, I am not allowed to consume butter any more. I miss that creamy flavor, especially in mashed potatoes. It used to be one of my favourites, but now I don't enjoy it as much. Is there any non-animal substitute for butter?
Vegan “butters” will be the closest substitute based on your requirements. They are specifically formulated to have the taste and mouthfeel of dairy butter, but contain no animal products.
Margarine, also a good substitute for dairy butter, is primarily vegetable based, but may contain a small amount of flavorings from dairy.
There are plenty of vegan 'butter' and margarine options out there, but these are primarily vegetable oil or soy-based. This only partially achieves your 'butter' goals. If it's the creaminess of butter you're after, I recommend trying something like Miyoko's European Style Cultured Vegan Butter. Its base is coconut and sunflower oil, but also includes cashew cream—a vegan fan favorite for achieving a creamy texture.
Most vegan butters are highly processed chemically. If you want to avoid any non-naturally occurring ingredients, an alternative for the kind of cooking you are talking about would be a coconut oil that is not chemically processed. If you use unprocessed coconut oil, you will have a strong coconut flavor in whatever it's added to.
Spectrum's organic oils (has to be its organic ones) that are refined are not chemically refined. They are mechanically refined. This would be something to try: https://www.spectrumorganics.com/product/organic-coconut-oil/ That should give you some creaminess. Flavor of butter is a different thing. But at least this version of a refined coconut oil will not bring tons of coconut flavor to the party.
To be clear, I'm not making any health claims at all. Mechanically refined coconut oil is not chemically refined with anything that could remotely be non-vegan or of any potential concern. There's no health claim. You have to research and decide for yourself on that front. I personally find the mechanically refined coconut oil tastes better. It's also organic. The chemically refined version is not.