Were the living standards in Medieval Europe a lot worse than in the Muslim world?

An indirect indicator is city size. Cities can only grow large when they, and the lands and trade routes that support them, remain well-protected long enough (and 50 years will do - cities can grow and decline very quickly). High living standards are hard to attain without cities.

Until 1800, the only cities under Christian rule with population exceeding 200,000 were Rome (300-400), Constantinople (500-1500), Paris (from 1300), Naples (from 1550), London (from 1600), Amsterdam (from 1700), Moscow and St. Petersburg (from 1750), Vienna (1790).

Meanwhile, during the rise of Islam (700-1000), Córdoba, Cairo, and Baghdad had grown to 500,000 or more - and subsequently declined in size. London and Paris only reached 500,000 by 1700.

After Córdoba fell to the Christians (1200), its population (which had already dwindled due to instability under Muslim rule) steadily decreased to 20,000; after Constantinople fell to the Ottomans, its population quickly increased from 200,000 to 500,000.

The general picture: larger cities (200,000) would grow in the Islamic world from 700 (but not so much after 1000), in Romance speaking countries from 1200, and in Germanic speaking countries only after 1550.

This nicely coincides with the term 'Renaissance', the "lifting of the Dark Ages", which, I think, stands for the cultural developments that followed from the economic rise in Europe (in Italy and Paris from 1200, the Netherlands and England from 1300). So while this is only indirect evidence, it suggests a positive answer to your question.

Angus Maddison provides some historical GDP data on his website.

From this data, at around 1000AD the per-capita GDP of Europe is about 425 1990 international dollars (this figure is about 31,000 for 2008 USA). The two stand-outs are Spain and Italy at 450, likely due to trade and contact with the Arab world.

Compare this figure to the Middle-Eastern average of 621. It's not a big difference, but economic disparities of that time would have been much lower than today - peasants were more or less equally poor, but to an observer living in those times, the difference would have been easy to spot.

It depends what period of Medieval History you're talking about (since the term can often be a catch-all referring to everything from late antiquity to the Renaissance, or can specifically refer to the period after the High Middle Ages - 14th/15th centuries), and what specific parts of Europe and the Middle East.

Note that Europe was severely depopulated at the time, intensive agriculture hadn't so much been "lost" as was the fact that lands were allowed to go out of cultivation and much of urban living had been whittled down to a few towns of 10k populaces each.

Luxury goods were certainly lost however, there was a catastrophic loss of skilled workers that went hand in hand with the decline of urban living and so on. The period most people think of when they're talking about the really "dark" part of the Middle Ages is essentially Central and Western Europe following the end of Justinian/Belisarius' abortive attempts to retake the Western portion of the Empire. The period after the actual fall (476) wasn't particularly bad, Theodoric was actually a fairly good ruler of Italy. But the 7th and 8th centuries saw chaotic mass internal migrations, Europe itself besieged on all sides by hostile existential threats (The Sassanids and then the Arab Caliphate) and so on.

A few notes on other things, tangentially related:

• European states, with their mix of either common and/or roman law were probably a fairer place to live. There are a couple of Arab sources contemporary to the Crusader Kingdoms that speak about the concept of what they call "Frankish justice" - Namely Ibn Juabyr and Ibn Munqidh. They both speak highly of it in comparison to the laws of the various surrounding Arab states. Shariah has always been kind of inefficient as a legal code for reasons I won't go into here.
• From about the Carolingian period (late 8th/9th centuries) onwards, Europe began a very slight recovery that accelerated in the High Middle Ages (11th to 14th). The real groundwork for the Renaissance was laid during this period. People like Gioto were, in some respects, the first "Renaissance painters".
• It's untrue that it was Muslims who preserved Greek texts, the Latin literary corpus had always survived largely intact in the West, but the only class who really made any use of it were Monastic Orders. What prompted the outpouring of literary growth in the Renaissance was just the increases in agricultural surpluses that led to increased urban growth and the flood of Greek scholars Westwards, who reunited the Greek literary corpus with the Latin for the first time since the loss of Rome after Justinian.
• A big part of the reason early middle ages science was so stagnant wasn't so much "Christian dogmatism" but an over-reliance on Aristotle.
• Byzantium, the remnants of Rome in the Greek East, had perhaps the best developed cities and standards of living of any Empire outside of the Tang and Song Dynasties in China until the defeat of the Byzantine Greeks to the Turks at Manzikert.
• Be wary of historical GDP estimates. India was wealthier than all of Europe combined until the early 19th century by these standards, purely because of the mass of grain it produced. Remember that most economic activity in those times were agricultural activities, and thus a place that produced a ton of grain was "rich". This didn't necessarily mean they were more technologically, culturally or socially advanced however.