"See Me Once" Overall Meaning:
GA could stand for "Google Analytics"
See me once, I'm watching your every move
Google analytics watches you through every website you visit.
Thanks for your visit, I know you were there
Again, google analytics tracks you once you visit a site by using cookies.
I see what you did there in my lair
The lair could mean "the internet", again google analytics tracking you.
Look for my acronym, and you'd have my proof.
And yeah, "GA" could stand for Google Analytics.
See me once, my lair is in the cloud
The cloud is another way of saying a web server, for hosting google analytics.
"See Me Twice" Overall Meaning:
I'm thinking of Lady Gaga, and the verb "to go gaga" over something
See me twice, almost glorious, a pretty cat
This is a stretch, but people can "go gaga" over something glorious by becoming "very enthusiastic and excited about someone or something". I'll be honest, I don't know about the "pretty cat".
thousands of reasons for hiding my hand
This is a huge stretch, but babies may go "gaga" (ga-ga-goo-goo), so you should hold onto their hand?
Maybe it's time for my outfit to be bland
Lady Gaga has many unique outfits that would be hard to call bland.
I can dive but not very deep, I guess I was born like that.
This is what got me on the idea of "Gaga". One of Lady Gaga's famous songs is "Born this way". Unfortunately, I'm not sure about "I can dive but not very deep", it could possibly have a musical meaning that I'm not aware of?
See me twice, find my songs in shroud
Again, a reference to music, which led me to the idea of Lady Gaga.
Edit: Added some additional info to the last line's partial explanation.