The achene is a type of simple dry fruit found on many flowering plants. The achene contains a single seed within it and is often mistaken as the seed itself. Other than strawberries, other achenes are the fruit of buttercup, buckwheat, and cannabis.
Some achenes have hair-like structures that make them appear like tumbleweeds. So, you aren’t getting annoying seeds stuck in your teeth when you eat a strawberry at all, but the fruit itself is stuck in your teeth. Too bad knowing this doesn’t make it any less annoying.