Got it, thanks…
Do you have a subscription?
Your profile says you’re using SketchUp 2018, which they stopped selling over two weeks ago.
If this is a new purchase, could it be that you got the $299 SketchUp Pro subscription? For subscription products there is no serial number, you just go to:
and sign into your account.
Or for the desktop app you would open it, and sign in then.
Your profile says SketchUp Pro 2018, which is no longer for sale, so can we assume you are trying to upgrade to 2019 (as opposed to that you lost the info for 2018)? If so, what version of license did you request? As @MikeWayzovski alluded, subscription licenses are managed by a Trimble server based on the id you use when you log into the desktop or web app; there is no serial number or license for subscriptions, only for the “classic” desktop license.
In addition, you would have received several emails.
Try your spam folder in the mailbox.