In Ancient Egypt the winged sun or Winged Disk it started probably as a Divine symbol , but later it became a Royal symbol , it meant the king him self and it was used by many ancient cultures worldwide ( with different meanings you can find it in bassicly in all civilizations of those days with all sorts of diferent meanings ).
As one of the most well renowned and respected Egyptologists Sir Alan Gardiner, who argued that the disk represented the ‘actual person’ of the of the king, (Dictionary of Ancient Egypt. Shaw & Nicholson, 2002, p305)
So Later it became a common symbol for kings and their families and no longer a Idol or Pagan specific identity
Like The seal of Tudhaliya IV used also the winged sun as a symbol of Royalty as many others in Egypt Babylon or Assyria
Probably it became the Symbol of King David Jewish House of Kings (like coat of arms)
So you find this Symbol in many seals Like the Solomon Seal in Jezebel Seal in Shebnaiah the servant of Uzziah and many others (all probably from the Kings Family )
In the begging Jewish seals had no drawings at all, later you find many of them with drawings with the symbol of their respected family:
Like the famous Yerovam Seal with a lion Or others with a boat or animals or flower etc..
The other Seal with the Beetle and the wings , it was the official Symbol of the Kingdom of Egypt in those days who Chizkiyahu was alied and was rebuked by Isaiah for doing so and trusting that they would come and help him against Sennacherib
As you cans see that Isaiah calls Egypt as "land of the beetle with wings" (Isaiah 18:1) and you can find in many places in Egypt these Symbol as the Symbol of the Kingdom of Egypt of those times
Therefore Chizkiyahu he probably used this Seal to Celebrate the Aliance with Egypt or to show trust etc.. With no Idol intentions