"Offline" means it's not connected to either cellular service or wifi, or the battery is dead, or the power is off.
If it says "Turn on Lost Mode..." then it's not turned on.
The store could NOT have turned on Lost Mode without you giving them the password to your girlfriend's iCloud account (They could have turned on
Find My iPhone in the iCloud preferences - that's a toggle.) The only way to put it into Lost Mode is to either use the Find My iPhone app, or go to iCloud.com. A repair shop has no reason to turn on Lost Mode in the first place.
(They might have asked you to turn
Find My iPhone off - Apple requires that before servicing any phone - they should have asked
you to do that by signing into iCloud.com yourself, so you could keep the iCloud password a secret.)
If they
had turned Lost Mode on, then you would now have the option to turn it off. Since you don't have that option...
If everything else is true, then the most logical answer is that they did not give you the same phone. Perhaps they did not put the broken screen back onto your phone, perhaps they gave you someone else's locked phone with a broken screen (one that someone else put into Lost Mode because it was lost or stolen, and never returned to them). You can use the IMEI/MEID number or Serial Number to identify the phone, depending on the model - see this article:
https://support.apple.com/HT204073