No, your tea does not contain soap! Tea leaves contain a molecule called saponin. It has soap-like surface active properties (Rai et al. 2021), so when the tea leaf releases saponin into water, you can see bubbles forming. I am not a physics expert, but briefly some parts of the saponin molecule either interact with water molecules or are repelled by water molecules and interact with air dissolved in the water. The saponin molecules will then exert a tension that will compress a stream of water that will encapsulate air.
In practice, some teas "foam" less than others. The later the tea is picked and the larger the leaf, the more saponin it contains! According to some researchers (Yu and He, 2018) One should not expect to see a spring Long Jing (Chinese green tea) "foamed" because it is less concentrated in saponin.
In chemistry, saponins are extracted by alkaline hydrolysis (Wang et al. 2015). Some tea drinkers would notice bubbles forming when using hard water (thus alkaline water). A saponin that would diffuse more depending on the water?
(sources on the french article https://sciencestea.wixsite.com/monsite/post/pourquoi-mon-th%C3%A9-mousse )