Does a multiple-entry Schengen visa allow visiting other Schengen countries later, without going via the issuing country?

You can enter any country within the Schengen area (25 countries) with no problems. See, for example, this unofficial Schengen visa website. I have done that personally before.

Anyway, out of personal experience, German immigration are not really friendly with this idea. Once in Frankfurt they refused the entry of a friend to the country because she had her visa issued from the Italian Embassy. Anyway after few tears they let her in. Moral of the story, Yes you can enter any Schengen country with your Schengen visa regardless of the issuing country.

It is not necessary for subsequent itineraries to include the country that issued the visa.

Your visa may be invalid for subsequent trips to the Schengen area, however. To be valid, it and you must meet the following conditions:

1. The visa's territorial validity must not be restricted to exclude any country you are visiting. To check this, look at the top of the visa where it says valid for; most visas are valid for the Schengen states, although they say this in the language of the country issuing the visa.

2. The visa must not expire before the end of your trip. Visas are issued for a limited period of time; the dates are indicated on the visa sticker with the headings from and until.

3. You must not have exhausted the visa's limit on entries. Visas may be issued for one, two, or multiple entries. If you have already used a single-entry visa once, or a dual-entry visa twice, you cannot use it again. The relevant heading is number of entries.

4. You must not have exhausted the allowed duration of stay. This is indicated under the heading duration of stay, and it is sometimes the source of confusion, so here are some points to consider:

   • The duration is given in days. Any part of a day counts, so if you enter at five minutes before midnight and leave five minutes after midnight then that counts as two days even though it was ten minutes.
   • If this number is 89 or less, then you must count the days of all visits cumulatively. For example, assume that you were given a 15-day duration of stay, that you were in the Schengen area for 8 days during your first trip, and that you arrive for your second trip on July 14th. In this case, you must leave the Schengen area on or before July 20th because you have seven days remaining out of the allowed 15.
   • if the number is 90 (it will never be more than 90), then you must follow the 90/180 rule. This means that for any period of 180 days, you are not allowed to spend more than 90 days in the Schengen area. If your visa is valid for longer than 180 days, then, you can spend more than 90 days in the Schengen area as long as you spread them out sufficiently to comply with the rule.