Twin reference in wrong order

You can fix the issue without changing the bst file (and making the copy editors happy, if you're submitting the paper and apalike is the required bib style). The sorting order, when the same list of author is shared by two entries, is determined by the title.

\begin{filecontents*}{\jobname.bib}
@inproceedings{landgren2016distributed,
  title={{\noopsort{A}{On}} distributed cooperative decision-making in multiarmed bandits},
  author={Landgren, Peter and Srivastava, Vaibhav and Leonard, Naomi Ehrich},
  booktitle={Control Conference (ECC), 2016 European},
  pages={243--248},
  year={2016},
  month=may,
  organization={IEEE}
}

@inproceedings{landgren2016bdistributed,
  title={{\noopsort{B}{Distributed}} cooperative decision-making in multiarmed bandits: Frequentist and Bayesian algorithms},
  author={Landgren, Peter and Srivastava, Vaibhav and Leonard, Naomi Ehrich},
  booktitle={Decision and Control (CDC), 2016 IEEE 55th Conference on},
  pages={167--172},
  year={2016},
  month=sep,
  organization={IEEE}
}
\end{filecontents*}

\documentclass{article}

\newcommand{\noopsort}[2]{#2}

\begin{document}

\nocite{*}
\bibliographystyle{apalike}
\bibliography{\jobname}

\end{document}

This way BibTeX sorts according to A and B, rather than according to On and Distributed.

I used filecontents* just for making the example self-contained.

With some modifications of apalike.bst you can sort by year and month. This will require you to input the three-letter month abbreviations without braces, though

month = may,

as is recommended in btxdoc anyway (p. 10)

month The month in which the work was published or, for an unpublished work, in which it was written. You should use the standard three-letter abbreviation, as described in Appendix B.1.3 of the LaTeXbook.

To modify apalike.bst proceed as follows.

1. Locate apalike.bst on your machine. You can do this by typing kpsewhich apalike.bst into the command line/terminal. Alternatively, obtain a copy of the file from CTAN http://mirrors.ctan.org/biblio/bibtex/base/apalike.bst

2. Copy the file to a location where TeX can find it. The document directory will do fine. See also https://texfaq.org/FAQ-inst-wlcf

3. Rename the file to apalike-month.bst (the license of apalike.bst requires you to change the name if you modify the file)

4. Apply the changes given in the following patch

   --- apalike.bst 2010-12-10 10:19:51.000000000 +0100
   +++ apalike-month.bst   2018-06-25 14:10:59.653275200 +0200
   @@ -1,7 +1,8 @@
   -% BibTeX `apalike' bibliography style (version 0.99a, 8-Dec-10), adapted from
   +% BibTeX `apalike-month' bibliography style
   +% based on `apalike' (version 0.99a, 8-Dec-10), which in turn was adapted from
    % the `alpha' style, version 0.99a; for BibTeX version 0.99a.
    %
   -% Copyright (C) 1988, 2010 Oren Patashnik.
   +% original copyright (C) 1988, 2010 Oren Patashnik.
    % Unlimited copying and redistribution of this file are permitted as long as
    % it is unmodified.  Modifications (and redistribution of modified versions)
    % are also permitted, but only if the resulting file is renamed.
   @@ -39,6 +40,7 @@
    %                    THIS `apalike' VERSION DOES NOT WORK WITH BIBTEX 0.98i.
    %    8-dec-10  (OP)  Still version 0.99a, as the code itself was unchanged;
    %                    this release clarified the license.
   +%   25-jun-18        Add month sorting
   
    ENTRY
      { address
   @@ -51,7 +53,7 @@
        institution
        journal
        key
   -%    month              not used in apalike
   +    month
        note
        number
        organization
   @@ -751,29 +753,29 @@
   
    FUNCTION {default.type} { misc }
   
   -MACRO {jan} {"January"}
   +MACRO {jan} {"01"}
   
   -MACRO {feb} {"February"}
   +MACRO {feb} {"02"}
   
   -MACRO {mar} {"March"}
   +MACRO {mar} {"03"}
   
   -MACRO {apr} {"April"}
   +MACRO {apr} {"04"}
   
   -MACRO {may} {"May"}
   +MACRO {may} {"05"}
   
   -MACRO {jun} {"June"}
   +MACRO {jun} {"06"}
   
   -MACRO {jul} {"July"}
   +MACRO {jul} {"07"}
   
   -MACRO {aug} {"August"}
   +MACRO {aug} {"08"}
   
   -MACRO {sep} {"September"}
   +MACRO {sep} {"09"}
   
   -MACRO {oct} {"October"}
   +MACRO {oct} {"10"}
   
   -MACRO {nov} {"November"}
   +MACRO {nov} {"11"}
   
   -MACRO {dec} {"December"}
   +MACRO {dec} {"12"}
   
    MACRO {acmcs} {"ACM Computing Surveys"}
   
   @@ -1013,6 +1015,13 @@
      *
      "    "
      *
   +  month empty$
   +    { "00" }
   +    { month }
   +  if$
   +  *
   +  "    "
   +  *
      title field.or.null
      sort.format.title
      *
   @@ -1072,6 +1081,13 @@
      *
      "    "
      *
   +  month empty$
   +    { "00" }
   +    { month }
   +  if$
   +  *
   +  "    "
   +  *
      title field.or.null
      sort.format.title
      *
   

   This patch re-enables the month field and includes the it field into the sorting sequence (if no month is given it is sorted as 00 so that work without explicit month sort before those with an explicit month, of course that can be changed by replacing the two "00"s with the desired sort string). To make sure that the months sort properly the three-letter macros are redefined to yield the month numbers. Note that the style has not been changed to print the month (so the sorting may feel somewhat arbitrary from the outside).

5. Use \bibliographystyle{apalike-month} instead of \bibliographystyle{apalike} in your document.

As alternative for steps 1 to 4 you can obtain the patched version of the file at https://gist.github.com/moewew/432f756b9fbb81d6337577cd78a8bbd8

With the new apalike-month.bst your MWE gives