Gdddk

The code below works fine

begintikzpicture
beginaxis
[ 
 ticks = none, axis lines = middle,
 xmin = -5, xmax = 5,
 ymin = -5, ymax = 5,
 ]
 foreach p in 1,2
 draw[red] (-5,1)--(5,1);
 
endaxis
endtikzpicture

However, as soon as I try to use the variable p inside the loop, as in

begintikzpicture
beginaxis
[ 
 ticks = none, axis lines = middle,
 xmin = -5, xmax = 5,
 ymin = -5, ymax = 5,
 ]
 foreach p in 1,2
 draw[red] (-5,p)--(5,p);
 
endaxis
endtikzpicture

I get an undefined control sequence error. This persists no matter what I use instead of p.

Welcome to TeX-SE! Notice that pgfplots does a lot of service in the background, which is why some variantes of foreach may not work in an axis environment. These things are explained in detail in section 8.1 Utility Commands of the pgfplots manual. Long story short for this case: use pgfplotsinvokeforeach instead.

documentclass[tikz,border=3.14mm]standalone
usepackagepgfplots
pgfplotssetcompat=1.16
begindocument
begintikzpicture
beginaxis
[ 
 ticks = none, axis lines = middle,
 xmin = -5, xmax = 5,
 ymin = -5, ymax = 5,
 ]
 pgfplotsinvokeforeach1,2
 draw[red] (-5,#1)--(5,#1);
 
endaxis
endtikzpicture
enddocument

There are situations in which you really want to use the traditional foreach loop. In those you can you can use this answer to get

documentclass[tikz,border=3.14mm]standalone
usepackagepgfplots
pgfplotssetcompat=1.16
begindocument
begintikzpicture
beginaxis
[ 
 ticks = none, axis lines = middle,
 xmin = -5, xmax = 5,
 ymin = -5, ymax = 5,
 ]
 foreach p in 1,2
 edeftempnoexpanddraw[red] (-5,p)--(5,p);
 temp
 
endaxis
endtikzpicture
enddocument

Side-remark: try to avoid calling the loop variable p, n, x or y if you consider making use of the calc library because there these macros have their own meanings.