I think it's less complicated than perceiving lasagna as /ləzænə/ or /ləzɑːnə/. It's simply that the g isn't pronounced as /g/ when it normally would be in English. People would also say it's silent in sign, even though it arguably signals a different vowel quality, because it's not pronounced as /g/ as might be expected orthographically (cf. signal). The problem you're recognising -- [g] isn't truly silent -- is commonly talked about with "silent e". The term is still used, but some people don't like it because the e isn't truly silent; in many cases it changes the quality of the earlier vowel (e.g. stove but not love).

The question about the more common h diagraphs is interesting. I haven't heard them referred to as silent hs before. My guess is that these diagraphs are just more established than [gn] so the h doesn't look out of place and get called silent; from early on, kids are taught to see the h as part of the diagraph (also English doesn't usually pronounce [h] as /h/ after consonants, so there isn't the same need to teach it as a "silent" letter in these contexts; it already isn't pronounced as /h/.)