If you have stable 50 Mbps throughput, that would be MORE than enough to stream even 4K video from Internet sources, but maybe it isn't stable.  Anyhow, the Dell Wireless 1506 is an old 802.11n card that only supports 2.4 GHz, and 50 Mbps is about the best you can expect in real-world throughput with a 2.4 GHz 802.11n connection, especially if you live in an area with a lot of nearby WiFi networks.  Given that you have a newer router and can get 270 Mbps from your phone, your best bet would be to upgrade your laptop's WiFi card to a model that supports 802.11ac and 5 GHz.  The Inspiron 3531 uses an mPCIe slot for its WiFi card, and in my opinion the best 802.11ac card made in that format before everything switched to the newer M.2 format was the Intel Wireless-AC 7260HMW.  You can find them online from various retailers for about $30.  I have one in my laptop and I can get my full 300 Mbps connection over WiFi, for what it's worth.

Installing the card would be easy.  The Service Manual for that system is here, and it includes step-by-step instructions complete with diagrams.  Just make sure you fasten the antenna connectors onto the card before you insert it into the slot, because that allows you to sort of pinch the connectors onto the card by having your thumb over the antenna connector and your index finger on the underside of the card beneath the connector.  It helps because a loose antenna connector can really kill your wireless performance.

Also make sure you get the Wireless-AC 7260 rather than the Wireless-N 7260 (no idea why Intel used the same model number for two completely different cards) and also the 7260HMW, not the 7260NGW.  The latter is the same card in M.2 format, which is the wrong slot type for your system.

And lastly, once you upgrade, make sure you connect to your router's 5 GHz network.  If your router uses different SSIDs for the two bands, then connect to the 5 GHz SSID.  If the two bands are named identically, then your system should automatically select the 5 GHz band at least when it's within reasonably good range of the router.

The 7260 card you bought is as good as it gets for your system, and it is certainly capable of more bandwidth. I can get my full 300 Mbps Internet connection on a 7260, and it would probably go higher if I had a faster connection. In terms of you only getting 150 Mbps, as mentioned above there are several factors. Is your smartphone consistently getting 270 Mbps in the same physical location that your laptop is consistently getting 150 Mbps? Don’t test them at exactly the same time of course. If so, make sure your laptop is connecting to 5 GHz if you use the same network name for both bands (which is recommended). To check, go to Settings > Network > WiFi, click the network you're connected to, and scroll down to check the information. If you’re on 5 GHz, make sure you securely connected both antennas to the card. I find it’s easiest to connect the antennas before inserting the card in the slot so you can “pinch” the antennas onto the card’s connectors with one finger on the underside of the card beneath the connector. If you still don’t see better performance, it could be the design of that system. If the antennas in your laptop are surrounded by aluminum rather than plastic that your smartphone might use, for example, that will reduce radio reception.