Schengen Visa - 2 Entries - 60 Days

I believe the duration of stay is per entry. I reach this conclusion because a multiple entry visa has number of entries "MULT" and duration of stay "90"; here the 90-day maximum is per entry, though for any given entry the actual number of days allowed might be reduced by the 90/180 rule. If I am correct about this, then you must leave the Schengen area on or before the 8th of June, the last day of the visa's period of validity.

Some countries' visas represent permission to enter the country (or to apply for entry), and it is possible to stay after the expiration date as long as you comply with the permitted duration of stay. In contrast, the Schengen visa allows you to remain in the Schengen area, and you must leave on or before the expiration date.

I do not know whether you can extend the visa in Sweden, but I doubt it. It may depend on the specific class of the visa, which you have not mentioned.

The Schengen regulations are somewhat unclear on this point. My understanding is that a visa like yours was never intended to exist. There are a lot of things in the Handbook for the processing of visa application and the modification of issued visas that suggest as much but the most important reason is perhaps two short provisions in annex VII of the Schengen Visa code.

4. ‘DURATION OF VISIT ... DAYS’ heading:

This heading indicates the number of days during which the visa holder may stay in the territory for which the visa is valid. This stay may be continuous or, depending on the number of days authorised, spread over several periods between the dates mentioned under 2, bearing in mind the number of entries authorised under 3.

[…]

When a visa is valid for more than six months, the duration of stays is 90 days in any 180-day period.

This also takes care of the issue because the 90-days-in-any-180-day-period business is only relevant for visas with a long validity and it's quite clear that in this case 90 days does not mean 90 days in total but 90 days per 180-day period. Everything else is supposed to be grand total of the number of days spent in the Schengen area.

That's why I suspect a long two-entry visa with less than 90 days of stay was never intended to exist by the people who drafted the rules. Consequently, it's difficult to decide how this limit ought to be interpreted.

Nowhere does the official documentation unambiguously state that it's OK to stay 60 days per entry. In fact, the duration of stay per entry never comes up as such. So it seems really risky to assume you can stay longer than 60 days in total.

In any case, extending your visa from within Sweden will be nearly impossible so the best thing to do is probably to get in touch with the consulate before leaving. See also What are the options for a non-EU national who wants to stay in Sweden for more than 90 days?