some plans and ideas for you
as inexpensive as the plastic totes are today, unless you just want to build something, for a toddler ( and a few years after that! ) I'd just use the clear plastic totes ... one for 'soft' toys, one for 'games', one for 'active' toys --- dolls, trucks, bats/balls, etc. ... and teach the child to sort/put away in the proper tote every evening before bed time. Everything has a place, and everything in it's place.
The prior suggestion is a good one - just make sure the lids won't lock and trap your child should he climb into it; you could look for sturdy cardboard boxes like from furniture or even the liquor store, cover them with wrapping or contact paper and let them serve temporarily until you are definitive about what you would like for your child. Good luck.
Decorate a cardboard box . Little ones like to play with the boxes the gifts came in. Hope you try it. Free.
Any suggestions for a 1 1/2 year old should probably not include wood. A heavy lid falling on tiny fingers is not a good idea. Decorate a plastic bin, make one out of a crate with no lid. Sand and paint their name or cover in material so there are no splinters.
They make hinges that prevent an accidently closing lid and trapping the head or a fatal blow.... https://www.amazon.com/s/ref=nb_sb_ss_i_2_17?url=search-alias%3Daps&field-keywords=hinges+for+toy+box+lid&sprefix=hinges+for+toy+bo%2Caps%2C260&crid=8AG5XY1XI1F1&rh=i%3Aaps%2Ck%3Ahinges+for+toy+box+lid
If you do build a box, make sure to buy the childproof hinges that help prevent smashed fingers.
Here is an idea using a plastic bin, paper and burlap or other fabric of your choice.