Tracing the origin of surname Singh | ਸਿੰਘ | सिंह among Sikhs, a thread:

First you must understand "casteism" is a social construct of Indian society not sanctioned by any religion, all Upper caste Hindus, Muslim( "Ashraf"), Sikh("Jatt", "Rajput") flaunt it.

Vedic Hinduism describes 4 "varnas":
•Brahmin [Religious Scholar]
•Kshatriya [Warrior]
•Vaishya [Farmer & Traders]
•Shudra [Labour]
A brahmin may become shudra by karma and vice versa.

As time passed the upper castes constructed social, administrative & economic institutions to make sure such system keeps their sons & descendants to continue the exploitation.

When Guru Nanak Dev Ji Maharaj was born he noticed this & decided to do away with the 4 varnas of Hindus & 4 sects of Muslims, as they had forgotten the true teachings of Vedas & Qur'an. Thus Sikhism was born.

Dasvi Patshahi Guru Gobind Singh Ji Maharaj decided to make the perfect Saint-Soldier, the Khalsa. ☬

A man with both shastra(religious scriptures) & Sashtra(weapon) vidya.
Kingship was bestowed upon us poor peasants

Coming back to original point, why "Singh" surname because the kshatriyas of this land had lost the fight to invaders. The Khalsa & Peshwas(brahmin) had to rise. Rajputs used Singh too but Kshatriya dharam is not just their birth right.

So it can be said to make Sikhs the upholder of righteousness, be on correct path of spirituality & to give equality the kes, kirpan & turban were mandated.

Kes, Kirpan(weapon)& Pagg(turban) used to be a sign of warrior notice it Sri Parshuram, Sri Krishna, Sri Ram, Prithviraj Chavan.
Krishna was Keshav "someone with good hair | slayer of asur késh"
Hindus should decolonize your mindset of hair & turban being a sign of backwardness.

Notice Chankaya now, incase you wanted to bring that everyone had kes.

Vishwaswaroop darshan was given to Arjun, a kshatriya to perform his duties not because he was born a kshatriya. Hindus seem to ignore this important aspect. You've been brainwashed with the myth of .
"Ahimsa".

Sikhs don't fall into your modern day definition of a Hindu but yes our Religion is born in BHARATVARSH | Hindostan & is an Indic non-brahmanic religion with Vedic sanction. So that's why it's put in the dharmic fold.

That's why our traditions are similar so is our bani with the use of Hari, Murari, Ram over anything else. For eg: Sri Krishna said "I'll be reborn every time injustice increases", Gurubani refers to Krishna as nirgun "almighty" or as sargun "form" in Dasam Granth among others.

So it is made clear it's one almighty/divinity we worship when we say the name of any god. But we do recognise god can nirgun("formless") & sargun ("swaroop").The sargun form of almighty is recognised in Chandi di var:

So, I hope I cleared a few doubts about usage of Singh as surname & Hindu Gods in popular culture among Sikhs. Not an expert, views are personal.

Open to critisism, just don't name call or try to troll. Won't respond.