I assume there's not a tempered glass screen protector on it? They come in 0.2mm and 0.3mm thicknesses typically and so stand off a fair bit. But I'd guess they'd tell you if they gave you one for free or something.
Otherwise yes, it should fit the same. If it isn't it's likely because the refurbisher (the company that provided the screens to your repair shop's supplier, or the repair shop itself if they did glass-only repairs) messed up. Either the frame wasn't glued properly (most likely) or they have too much LOCA (liquid glue, which they shouldn't be using anyway) or a double OCA (sheet glue) layer. I can guarantee you that they didn't put extra thick glass on the screen.
Bring it back. All of the major parts suppliers will RMA faulty parts and give them a new one so it won't cost them anything other than time. And they should have seen the problem before they started.
I have to change my answer as I recently learned some info that relates to this issue:
There have been a spate of iPhone 6 replacement screen issues as of late, the most common of which is screen to frame separation. The problem is that the design of the iPhone 6 leaves very little area of contact between the frame and the glass. Original Apple screens don't have a problem with this as they use a special type of glue that is held in a press while it sets, which can take hours. Aftermarket manufacturers don't want to deal with that so they use hot glue. But the type of hot glue they were using they found ineffective at holding the frames on, so to increase strength they made the frames thicker. These thicker frames allowed greater glue contact (in theory), but they caused the LCD to protrude past the original. Most manufacturers have since solved this issue by switching to a stronger glue.
As it turns out, this didn't work either, and now there have been numerous failures. So if you find your screen separates from the rest of the phone bring it back and get it replaced. In fact, if you call your installer they may be able to swap yours out for a replacement as their supplier (if they're using a reputable supplier) will likely issue an RMA without any problems for the old screens.
tldr; it's not an installer fault, it's a supplier issue.