Jelly, 24 bytes

FLḶ0ẋ;þµ^þ/ẎŒḊ=3$$¡=1Ạ€Ẹ

Try it online!

-1 byte thanks to Mr. Xcoder

Another solution which I developed completely independently of Jonathan Allan but now looking at his is exactly the same:

Jelly, 16 bytes

FLḶ0ẋ;þµŒpSP$€1e

Try it online!

Jelly, 16 bytes

FLḶ0ẋ;þµŒpSP$€1e

Try it online!

-1 byte thanks to Mr. Xcoder

I developed this completely independently of Jonathan Allan but now looking at his is exactly the same:

Jelly, 24 bytes

FLḶ0ẋ;þµ^þ/ẎŒḊ=3$$¡=1Ạ€Ẹ

Try it online!

-1 byte thanks to Mr. Xcoder

Jelly, 24 bytes

FLḶ0ẋ;þµ^þ/ẎŒḊ=3$$¡=1Ạ€Ẹ

Try it online!

-1 byte thanks to Mr. Xcoder

Another solution which I developed completely independently of Jonathan Allan but now looking at his is exactly the same:

Jelly, 16 bytes

FLḶ0ẋ;þµŒpSP$€1e

Try it online!

Jelly, 15 bytes

FLḶ0ẋ;þµ^þ/ẎẠ€Ẹ

Try it online!

Explanation

FLḶ0ẋ;þµ^þ/ẎẠ€Ẹ  Main Link
FL               Take the length of all lists flattened (sum of lengths)
  Ḷ              0 .. sum (offsets)
   0ẋ            Repeat `[0]` by each of the offset sizes
     ;þ          Product table by `;` (gets `n` lists of lists for input size `n`, with each sublist containing all shifts of the original lists)
       µ         <new monadic chain>
          /      Reduce:
        ^þ       Product table of XOR
           Ẏ     Tighten; contains all attempted tilings now
            Ạ€   All elements truthy for each sublist?
              Ẹ  Any sublist had all 1s?

For whatever reason, you need to input the falsy test case [1, 0, 1] as [[[1, 0, 1]]]; I think that's because of the way Jelly processes input.

-1 byte thanks to Mr. Xcoder

Jelly, 24 bytes

FLḶ0ẋ;þµ^þ/ẎŒḊ=3$$¡=1Ạ€Ẹ

Try it online!

-1 byte thanks to Mr. Xcoder