Commodore C64, using PETSCII graphics, ~392 BASIC bytes

The simplest way to do this is to use the reverse character mode to print out white spaces, but changing the colours as necessary. These are Control characters in Commodore BASIC, similar to escaped characters in a C or PHP string. We will use a string array B$, and then a FOR/NEXT loop to output to the C64 screen. All colours in this listing are accessed by the control (often referred to as CTRL) key, by pressing the following key combinations:

{BLACK} is CTRL + 1
{WHITE} is CRTL + 2
{CYAN} is CTRL + 4
{BLUE} is CRTL + 7
{REVERSE ON} is CTRL + 9

To make the listing clearer and easier to type in, {SPACE x 5} indicates 5 spaces following the control character, as looking at the listing in Commodore BASIC might not be very clear.

Here is the listing:

0DIMB$(12):B$(.)="{BLACK}{SPACE x 5}{CYAN}{SPACE x 4}{BLACK}{SPACE x 5}":B$(1)="{BLACK}{SPACE x 3}{CYAN}{SPACE x 8}{BLACK}{SPACE x 3}"
1B$(2)="{BLACK}{SPACE x 2}{CYAN}{SPACE x 10}{BLACK}{SPACE x 2}":B$(3)="{BLACK}{SPACE}{CYAN}{SPACE}{WHITE}{SPACE x 2}{CYAN}{SPACE x 4}{WHITE}{SPACE x 2}{CYAN}{SPACE x 3}{BLACK}{SPACE}"
2B$(4)="{BLACK}{SPACE}{WHITE}{SPACE x 4}{CYAN}{SPACE x 2}{WHITE}{SPACE x 4}{CYAN}{SPACE x 2}{BLACK}{SPACE}":B$(5)="{BLACK}{SPACE}{BLUE}{SPACE x 2}{WHITE}{SPACE x 2}{CYAN}{SPACE x 2}{BLUE}{SPACE x 2}{WHITE}{SPACE x 2}{CYAN}{SPACE x 2}{BLACK}{SPACE}"
3B$(6)="{CYAN}{SPACE}{BLUE}{SPACE x 2}{WHITE}{SPACE x 2}{CYAN}{SPACE x 2}{BLUE}{SPACE x 2}{WHITE}{SPACE x 2}{CYAN}{SPACE x 3}":B$(7)="{CYAN}{SPACE x 2}{WHITE}{SPACE x 2}{CYAN}{SPACE x 4}{WHITE}{SPACE x 2}{CYAN}{SPACE x 4}"
4B$(8)="{CYAN}{SPACE x 14}":B$(9}=B$(8):B$(10)=B$(8):B$(11)="{CYAN}{SPACE x 2}{BLACK}{SPACE}{CYAN}{SPACE x 3}{BLACK}{SPACE x 2}{CYAN}{SPACE x 3}{BLACK}{SPACE}{CYAN}{SPACE x 2}"
5B$(12)="{CYAN}{SPACE}{BLACK}{SPACE x 3}{CYAN}{SPACE x 2}{BLACK}{SPACE x 2}{CYAN}{SPACE x 2}{BLACK}{SPACE x 3}{CYAN}{SPACE}":FORI=.TO12:PRINT"{REVERSE ON}"B$(I):NEXT

Some notes: be careful when entering line 4 as it is 80 characters in length (the maximum number of characters in a logical C64 line). Cursor up so that the flashing cursor is over one of the characters on the line and press the return key. Or, enter this into a Commodore C128, which allows for 160 characters in a logical line; this will work in C64 or C128 mode. I typed this in again from the listing above to check that the number of spaces and control characters are correct, and it seems fine. Best viewed with POKE 53280, 0: POKE 53281, 0: RUN.

By using C128 BASIC, and with the 160 PETSCII character per logical line, this can be crunched further as you will be able to reduce the line numbers. Some bytes could also be saved by removing some of the control characters, for instance, where B$(0) ends with {BLACK} there is no need to start B$(1) with {BLACK}. I will try to crunch this further to save some more bytes.

And here is how things look on a C64 (at least a C64 emulator):

C (gcc), 151 bytes

-3 thanks to @vengy
-28 thanks to @ceilingcat

Essentially a port of my JS answer.

n;y;main(x){for(;n<195;printf(x>13?"\n":"[4%cm ","6074"[L"㸟㠇〃  ↇƆ\0\0\0\0ࣄ᳎"[y=n++/15]>>x&1|(y>2&y<8)*7800/(~y&3?2:5)>>x&2]))x=n%15;}

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Output

C (gcc), 147 bytes

i=182;main(){for(;i--;)printf("\e[4%cm %s","0674"[3&"ATPATQETQUUUUUUUUUUUUUUUeVeVUzWzGe~e~PiZiJTiUiATUUE@TUU@@PU"[i/3]>>i%3*2],i%14?"":"\e[0m\n");}

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-4B from ceilingcat

Port like

Assumes that after ...TUU@@PU there are 2 null bytes

Perl 5 + -p0513, 132 bytes

s!!5.2B8.4B5.5B3.CLBCLB2.2MB2MB2.,MB,MB.@/L@/L2BM2BM24B.B<.B<B<2.B2.B3..!;s/\d+(.)/$1x$&/ge;s/./ord$&/ge;s/.{14}\K/
/g;s/./.[4$&m /g

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Explanation

The string of random numbers and symbols is an RLE string that contains the data for all the background colours (only the last digit, as the first digit's always 4), this is stored in $_ via the s///ubstitution operator. This is decoded to repeated runs of characters and each of those characters is replaced with its ordinal value, split into run of 14 characters and has the ANSI escape code string wrapped around each value.

JavaScript (Node.js), 150 bytes

f=(x=182)=>x?`\x1B[4${'0674'[3&Buffer('ATPATQETQUUUUUUUUUUUUUUUeVeVUzWzGe~e~PiZiJTiUiATUUE@TUU@@PU')[--x/3|0]>>x%3*2]}m `+(x%14?'':`\x1b[0m
`)+f(x):''

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Hardcode

Arnauld's solution spill color and mine don't. Removing this difference mine is shorter, but Arnauld usen't Buffer

I don't use char \x1B since otherwise it's better encode 4 pos in a char

C (gcc, msvc), 228 bytes

#include<stdio.h>
j,i[]={89391445,88080405,83886085,77597185,111684225,132660865,65552000,10488320,0,0,0,4214800,22040660};main(){for(;j<182;j++)printf("\x1B[4%cm%s","6074"[(i[j/14]>>(2*(13-j%14)))&3],j%14==13?" \x1b[0m\n":" ");}

NB: The actual code contains the unprintable ESC character \x1b.

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Output

Windows Command Prompt:

Linux (Ubuntu) Terminal: