Hello @Anonymous,
Max is a bit different than Blender so there are some differences. Technically what you are showing in blender is incremental move. I think it would be cool if Max's worked the way you showed in the 2nd movie, but it currently doesn't. Max handles incremental move by right clicking on the move icon and typing a value in the "Offset" column. This would make a good Feature Request, and if you do make one and post the link here I'll vote on it.
A snapping half-solution would be the following: There are many options in the snap settings (right click any snap icon to bring up the settings). In your case, you probably won't be able to get that behavior on the perspective view, but you could in ortho
If you open the snap settings and float it and get something just like this shown below. Check "Enable Axis Constraints" as shown below, then use Grid Points or Grid Lines as snaps. If you have setup your grid to the size you want them to snap, you can snap onto those grid points easily in ortho.

That way requires you to use ortho views which means the grids already exist. You can technically create your own grids in Max but it's clunky enough in perspective that I don't think it's worth the hassle for what you are trying to achieve.
The best solution would likely be a script that does an incremental move. You may be able to find one online or ask in the Programming forum to see if someone would make one.
Best Regards,