Can angels be photographed? Is it possible they have a shape similar to humans?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

Angels are creatures of Allah just like any other creatures. Allah has created them from light, and He gave them the possibility of taking different shapes. Indeed, Jibreel (Gabriel) used to come to the Prophet in the shape of the known Companion whose name is Dihya al-Kalbi .

Al-Bukhaari and Muslim reported that Jibreel came to the Prophet while Um Salamah was present. Jibreel started talking to the Prophet and then left. The Prophet asked Um Salamah, “Do you know who it was?” (or a similar question). She replied: “It was Dihya.” (a handsome person amongst the companions of the Prophet). Later on Um Salamah said: “I swear by Allah! I thought he was none but Dihya, till I heard the Prophet talking about Jibreel in his sermon.”

Besides, Imaam Ahmad reported that a man from the Ansar said that he saw the Prophet talking for a long time with a man. After he left, the man from the Ansar came and asked the Prophet about that man. The Prophet asked him: “Did you see him?” He replied “Yes”. The Prophet asked: “Do you know who he was?” The man replied: “No”. The Prophet said: “That was Jibreel, he kept on advising me about the neighbor till I thought that he was going to give him a share in the inheritance.” The Prophet added: “Had you greeted him with Salaam, he would have replied to your greeting.”

As for taking the picture of the angel with a camera, there is nothing in the Quran or the Sunnah that implies the possibility or impossibility of doing so.

The correct opinion is that if they come in the form of human beings, they could be photographed, otherwise they can't.
However, even if it is possible to take pictures of them, it is forbidden to do so as they are live beings and there is sound evidence that taking pictures of living beings is not permissible unless there is a necessity or a dire need; such conditions are not present as far as Angels are concerned.

Allah knows best.