Notice that $2021$ is a concatenation of consecutive integers: $20\sim 21$
Also $2021$ is a product of consecutive primes: $43\times 47$.

What is the next number with both of these properties?
$24502451$ is close, $4943\times 4957$ but $4951$ is in between.
$2484224843$ is close, $49831\times 49853$ but $49843$ is in between.
$715353612\sim 715353607$ isn't quite there but is $845785793\times 845785799$

With Mathematica, we can find more solutions.
794018604377235322848433897872605582 ~
794018604377235322848433897872605583 =
891077215721081784886888257701070827 ×
891077215721081784886888257701070829

2518711810848159770018909254809359591672377471484881441744436703324716 ~
2518711810848159770018909254809359591672377471484881441744436703324717 =
5018676928083894672666012088036109843105301546773725102790665815794437 ×
5018676928083894672666012088036109843105301546773725102790665815794441

353879205744237011544616255111782082608671961515039134082358165448687146 ~
353879205744237011544616255111782082608671961515039134082358165448687147 =
594877471202462845078583461328011525336167267541426222873827376039101347 ×
594877471202462845078583461328011525336167267541426222873827376039101401

Some code to use is the following:

modPass[k_] := Abs[Mod[k, 6, -1]] == 1;
solve2021[exp_,add_]:=
Module[{cases},
cases=Solve[{10^exp  a +(a+1) == b (b+add),10^(exp-1)< b<10^exp}, Integers];
If[Length[cases]>0,
Select[Select[b/.cases,modPass[#] && modPass[#+add]&], PrimeQ[#] && PrimeQ[#+add]&]]]

solve2021[36, 2]
{891077215721081784886888257701070827}
solve2021[70, 4]
{5018676928083894672666012088036109843105301546773725102790665815794437}
solve2021[72, 54]
{594877471202462845078583461328011525336167267541426222873827376039101347}

Still a bit more to check, there may be intervening primes. For example, solve2021[19, 76] has intervening primes.

What are other solutions?