The prayer pose you recognize is a sign of respect and greeting.
This is purely your own interpretation though (and an incredibly selective one). If a Catholic priest walked up to you and stood with legs together and hands clasped in exactly the same yoga pose, you'd assume it was religious in nature, right? But a yoga instructor does the same thing, and it's not? Why? Doubly so when the poses were created specifically as part of prayer rituals for a real religion. It's no different. You've just arbitrarily decided that eastern spiritualism doesn't count as a 'real religion" within the context of US separation of Church and State, while western faiths (like Catholicism) do count.
It's arbitrary and purely within your own mind. There is nothing more or less inherently religious about saying 5 Hail Mary's versus performing a yoga series like the Sun Salutation. Nothing at all. The fact that one prayer takes the form of a series of physical actions designed to bring you closer to your spiritual goals, while the other is a chanted series of words designed to do the same really shouldn't matter at all.