Can I remove my PS3 HDD and put it back again?

Yes, you can. You should not have any format issues, either.

You will often hear of users having to reformat hard drives for their console, in cases where they are installing a new hard drive. Different devices understand different formats, and in most cases, consoles accept different formats to regular computers. For this reason, portable hard drives do not come with standard console format, as they are designed for computer use. The console has to first format the drive, to convert it to a format the console understands.

Given that the hard drive not only came from a PlayStation, but the same PlayStation, nonetheless; You should have absolutely no problems what so ever.

AFAIK, if you were to connect to a different PlayStation, you would need to reconfirm your user account password. Apart from that, the drive should work on other PlayStation consoles with no trouble, too.

The answer from user106385 is absolutely right, and I marked it as the accepted answer. But I want to add some information.

In the PS3 Super Slim (my case) there's a thing that might confuse the user. When you insert the same HDD after removing it, the console asks you two questions: if you want to change the storage location to the HDD, and if you want to move the internal flash data to the HDD.

Well, you have to answer yes to the first question and no to the second. Logical. But I though that by answering no to the second, I would not be able to do what I wanted. Glad it worked, and at the end, I got all the data.