Is it possible to replace a digit with the next digit using regex?

To add 1 to the number after the . on each line using any awk in any shell on every Unix box would be:

awk 'BEGIN{FS=OFS="."} {++$2} 1' file

When considering whether or not to try to use regular expressions to solve any given problem, always keep in mind this quote:

Some people, when confronted with a problem, think “I know, I’ll use regular expressions.” Now they have two problems.

This might work for you (GNU sed):

sed -E '/^1\.1[0-9]/s/(...)(.)(.*)/echo "\1$((\2+1))\3"/e' file

If a line begins 1.1 followed by a digit, replace that line with an echo command that when evaluated adds 1 to the fourth character.

Or if you prefer:

sed -E 's/^(1\.1)([0-9])(.*)/echo "\1$((\2+1))\3"/e' file

Use this Perl one-liner:

perl -i.bak -lpe 's{^(\d+[.])(\d+)}{$1 . ($2 + 1) }e' input_file

The Perl one-liner uses these command line flags:
-e : Tells Perl to look for code in-line, instead of in a file.
-p : Loop over the input one line at a time, assigning it to $_ by default. Add print $_ after each loop iteration.
-l : Strip the input line separator ("\n" on *NIX by default) before executing the code in-line, and append it when printing.
-i.bak : Edit input files in-place (overwrite the input file). Before overwriting, save a backup copy of the original file by appending to its name the extension .bak. If you want to skip writing a backup file, just use -i and skip the extension.

s{PATTERN}{REPLACEMENT} : Replace regex PATTERN with REPLACEMENT.
^ : Matches the beginning of the line.
(\d+[.]) : Matches 1 or more digits, followed by a literal dot (.). Parentheses make it store the match in capture variable $1.
$2 + 1: The string stored in capture variable $2, evaluated as a number, plus 1.
$1 . ($2 + 1): Concatenates $1 and the result of $2 + 1, now as strings.

The regex uses this modifier:
e : Evaluate REPLACEMENT as an expression in s{PATTERN}{REPLACEMENT}.

See also:

• perldoc perlrun: how to execute the Perl interpreter: command line switches
• perldoc perlre: Perl regular expressions (regexes)
• perldoc perlre: Perl regular expressions (regexes): Quantifiers; Character Classes and other Special Escapes; Assertions; Capture groups
• perldoc perlrequick: Perl regular expressions quick start

Is there such a feature in any regex engine/implementation that can do this sort of transformation of captured groups?

Python's re.sub does accept function as 2nd argument, which will be furnished with Match object. Consider following example, let subexample.py content be

import re
text = "1.10...\n1.11...\n1.12...\n1.13...\n...\n1.18...\n"
def replacement(m):
    value = int(m.group(1))
    value += 1
    return "1.1%d" % value
result = re.sub(r"1[.]1([0-9])", replacement, text)
print(result)

then running python subexample.py will give following output

1.11...
1.12...
1.13...
1.14...
...
1.19...

Explanation: I prepared function replacement function which does access 1st (and only) group, convert it to numeric value (integer), increase by 1 and then prepare replacement string using % formatting. Note that [.] was used in regular expression to mean literal dot, not any character.

(tested in Python 3.12.3)