Is it possible to realistically use Lydia for the entire game?

Yes, you may absolutely use a strong and powerful Lydia for the entire game. I'll explain below.

Problem

As mentioned in the other answers here, the game locks followers around the level you are when you acquire them. Because of Lydia's typically early acquisition in the game, she is leveled extremely low and will become useless if nothing is done while you become stronger.

EDIT: As of patch 1.6 this problem has been fixed. However I am not positive if the fix affects current saves, so this solution may still need to be used for your currently under-leveled followers from games started pre-1.6

Solution

The Wabbajack - A staff granted for completing the Daedric Quest: The Mind of Madness

The quest is extremely easy and doable at any level. You must have access to Solitude. I'll let you do your own research on the quest itself. To the point:

• Important: Release Lydia from your service first.
• Use the Wabbajack on Lydia until she morphs into a another creature: A goat, bunny, daedra, whatever. It's important that she morphs.
• Use the Wabbajack again to morph her back. If she returns wearing her default armor, the attempt has been successful and she will be re-leveled to your current level.
• Re-hire Lydia as your follower.

Why? The act of morphing her to another creature, and then morphing her back resets her stats to your level. Yay.

Notes

• Defintely quick save or save before attempting to use Wabbajack on Lydia. The wabbajack is random and has many effects including Instant Death.
• Take any rare items from her before the transformation attempt, as she sometimes loses them on her return to a human state

That's it. I've put in over 200 hours in Skyrim and I've had Lydia the entire time. Currently on Master @ level 55 and Lydia is basically her own demi-god. Have fun.

While I can't find specific string in patch notes, many users reported and UESP wiki notes that followers level as intended post patch 1.6.

In case you still have problems with old followers or somehow unable to upgrade, you can either regenerate follower with console commands on PC or use wabbajack on other platforms to make follower transform to creature and then back, thus re-rolling their stats (this will be noted by follower reverting to their original gear). Stats will be updated to their current level.

No, followers won't level with you (see this question). This means she will rapidly lose effectiveness at combat. You can still take her with you, but she will be crawling around on the ground most of the time simply due to her low level.

Instead, if you purchased the Whiterun house, you can just leave her there as a "second wife". Just dismiss her and she will wait in your house until you return.