Are many of the traditionally oil dependent Gulf states really successfully diversifying their economies? I've often heard the claim that many Persian Gulf states are successfully diversifying their economies away from oil (e.g. non-oil share of UAE's GDP is 71% and in Saudi Arabia, non-petroleum sectors account for 55% of GDP).

The IMF has the 2015 GDPs per capita of the UAE and Saudi Arabia at \$35392 and \$20139, respectively. If we naively multiply by 0.71 and 0.55, we get that their non-oil GDPs per capita are \$25128 and \$11076. \$25128 is greater than the GDPs per capita of countries such as Portugal and Poland, and \$11076 is greater than those of Mexico and Turkey.

I'm rather surprised by this, as I've always assumed that countries such as Portugal and Mexico would be well ahead of the UAE/Saudi Arabia in terms of development/education/technology etc.

Are the 71% and 55% figures (and similar claims for other Gulf states) misleading in some way? 

i.e. Are most of the non-oil sectors in these countries still somehow mostly based on oil (and would they substantially shrink/collapse if the oil "ran out")?
Several points need to be considered.
1) Oil dependance
First of all, it is not correct to assert that 30%-50% of the GDP being concentrated within a single industry is &quot;diversified&quot;. This is actually a high focus. And I can only guess that a lot of other activities are strongly linked to the oil and gas sector: for example, banks might have a lot of lending and trading activity in the sector, even if they aren't directly counted as being part of it.
Moreover, an interesting thing to look at is how much the government relies on this sector. For Saudi Arabia, which budget was discussed recently, it is obvious that oil plays a major role, as it makes up 90% of the revenue. FOr the UAE the answer is more tricky, as the different emirates have significantly different profiles: Abu Dhabi relies strongly on the Oil industry, whereas the other provinces such as Dubai rely only slightely on oil - though they receive high subsidies from Abu Dhabi.
As for the progress made, there are indeed some. These countries are aware that they will not always be able to pump up oil, and that they will need, in the long run, to have other strong industries. That is why, pretty much the same as in Norway, huge sovereign funds (check nb 4 &amp; 6) are setted up in these countries.
2) Development

I'm rather surprised by this, as I've always assumed that countries
such as Portugal and Mexico would be well ahead of the UAE/Saudi
Arabia in terms of development/education/technology etc.

GDP per capita is not a index of development. If you are looking for this, the HDI (Human development indicator) is a much more appropriate index. This index is composed by several measures of education, wealth (GDP), life expectancy and other indicators which relate to development and not only to economics.
The UN compiles this indicator on an annual basis, here is for 2015. It shows the following for the countries you mentioned. The HDI ranges from 0 (worse) to 1 (perfect score):

#6 Ireland: 0.916
#36 Poland: 0.843
#39 Saudi Arabia: 0.837
#41 United Arab Emirates: 0.835
#43 Portugal: 0.830
#72 Turkey: 0.751
#74 Mexico: 0.756

You can see from this numbers that the gap is thin between Portugal and the two Oil countries. You can also guess that the contribution of their GDP to the index is much higher than for any other countries mentionned above. Thus, the pure non-economic indicators are probably better for the other countries than Saudi Arabia and the UAE.
Furthermore, the Gini index can also be used to have some idea of the level of development of a country. It shows how big inequality are (regarding income or net wealth for example). Sadly this index is not computed for Saudi Arabia nor for the UAE. This is probably intentional, as these countries are well known to have very high inequality, in particular between immigrants (the huge majority of UAE's inhabitants) and natives.
Finaly, your intuition is correct: it is indeed quite 'naive' to multiply the GDP per capita by the proportion of GDP of a sector. Indeed, different sectors display significantly different productivity and efficiency of the labor factor. Therefore, it can be that the productivity of the Oil industry in those countries largely outweigh the productivity of the other sectors. Therefore it is hard to compare this clumsy calculation with 'full' GDPs of other countries.