Does Steam transfer offline save files from one device to another? (specifically Undertale)

Solution 1:

There are basically 3 way that games on Steam can track your progress between PCs.

Steam Cloud

If a game has been made to use this feature, Steam games can store their saves to Steam's save cloud. Steam will then make sure that the use will have their save data remain in sync between different PCs. You can see if a game uses Steam Cloud by checking its store page, or your Library in List View mode if you already own the game.
Undertale does use Steam Cloud, despite the feature not appearing on the store page.

Steam Achievements/Leaderboards

This is quite rare, but some games use Steam Achievements or Leaderboards to track how far you've made into the game. There are only a few cases where this is even viable, but I know at least one example that opts to use this in stead of other syncing methods.

Third Party Storage

Most often seen on online-only games, like MMOs, some games track your progress by tying it to an external account. You'll have to log into this account separately, because this is the only way for the third party server to identify you. Although it may be possible that they could identify you with your Steam account, using your Steam account as a log-in substitute.

Solution 2:

Undertale uses the Steam Cloud to sync saves between machines.

However, you can disable Steam Cloud for individual games.

If you enter the Properties panel for a game in your library (available on the right-click menu on PC), and go to the Updates tab, any game that has Steam Cloud functionality will provide a checkbox that you can uncheck in order to disable the synchronization.

Undertale normally syncs game state data between machines, including your save file. If you disable the Steam Cloud option for the game before installing it on the second machine, none of this data will be transferred over, and both machines can have separate experiences with the game.

Solution 3:

As the other answers point out, Undertale uses the Steam Cloud for saves which will work against what you're trying to accomplish. However, I wouldn't recommend disabling Steam Cloud to resolve this problem. Instead, what you should do is create separate Steam accounts for both you and your brother. One of you should buy Undertale, and then use Steam Family Sharing to share the game to the other account. You won't be able to play the game at the same time, but you can't log into the same account at the same time either.

This way Steam Cloud becomes advantage rather than a disadvantage. For example, if it ever become to be necessary or convenient you can log on to your account using your brother's computer and continue your game of Undertale without interfering with his. Also if things like achievements or leaderboards are ever added to the game, they'll be automatically tracked separately. Perhaps, more importantly doing this way is perfectly legal. What you're suggesting doing, sharing an account between two people, is not permitted according the Steam Subscriber Agreement.