You can spatial join the dataframe to itself, intersect, the output will be the lines where two polygons share a border, and measure length:
import geopandas as gpd
import numpy as np
df = gpd.read_file(r"/home/bera/Desktop/GIStest/shared_length.shp")
# id geometry
# 0 1 POLYGON ((594912.796 7238961.152, 596731.160 7...
# 1 2 POLYGON ((594912.796 7238961.152, 595300.862 7...
# 2 3 POLYGON ((595300.862 7239216.167, 594912.796 7...
# 3 4 POLYGON ((598039.495 7240524.502, 598294.510 7...
df["geombackup"] = df.geometry
sj = gpd.sjoin(df[["id","geometry"]], df[["id","geombackup","geometry"]], how="left", predicate="intersects")
# sj[["id_left","id_right","geombackup"]].head()
# id_left id_right geombackup
# 0 1 1 POLYGON ((594912.796 7238961.152, 596731.160 7...
# 0 1 2 POLYGON ((594912.796 7238961.152, 595300.862 7...
# 0 1 3 POLYGON ((595300.862 7239216.167, 594912.796 7...
# 1 2 1 POLYGON ((594912.796 7238961.152, 596731.160 7...
# 1 2 2 POLYGON ((594912.796 7238961.152, 595300.862 7...
#Drop self joins, like the first row above
sj = sj.loc[sj["id_left"]!=sj["id_right"]]
#Drop duplicates, like the second row is the same as 4th. 1-2 and 2-1
sj = sj.loc[~sj.geombackup.duplicated()]
sj["borderline"] = sj.apply(lambda x: x["geometry"].intersection(x["geombackup"]), axis=1)
# sj[["id_left","id_right","borderline"]].head(1)
# id_left id_right borderline
# 0 1 2 LINESTRING (594912.796 7238961.152, 596731.160...
sj["borderlength"] = sj["borderline"].length
sj = sj.set_geometry(col="borderline", crs=df.crs).drop(columns=["geometry","geombackup"])
sj.to_file(r"/home/bera/Desktop/GIStest/shared_borders.shp")
